Joint Optimization of the Spatial and the Temporal Discretization Scheme for Accurate Computation of Acoustic Problems

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1 Commun. Comput. Phys. doi:.428/cicp.oa-27-9 Vol., No., pp Joint Optimization of the Spatial and the Temporal Discretization Scheme for Accurate Computation of Acoustic Problems Jitenjaya Pradhan, Saksham Jindal 2, Bikash Mahato and Yogesh G. Bhumkar, Scientific Computing Laboratory, School of Mechanical Sciences, IIT Bhubaneswar, Khurda 7525, Odisha, India. 2 Department of Ocean Engineering and Naval Architecture, IIT Kharagpur 7232, West-Bengal, India. Received 7 September 27; Accepted (in revised version) 24 December 27 Abstract. Here, a physical dispersion relation preserving(drp) scheme has been developed by combined optimization of the spatial and the multi-stage temporal discretization scheme to solve acoustics problems accurately. The coupled compact difference scheme (CCS) has been spectrally optimized (OCCS) for accurate evaluation of the spatial derivative terms. Net, the combination of the OCCS scheme and the five stage Runge-Kutta time integration(ork5) scheme has been optimized to reduce numerical diffusion and dispersion error significantly. Proposed OCCS ORK5 scheme provides accurate solutions at considerably higher CFL number. In addition, ORK5 time integration scheme consists of low storage formulation and requires less memory as compared to the traditional Runge-Kutta schemes. Solutions of the model problems involving propagation, reflection and diffraction of acoustic waves have been obtained to demonstrate the accuracy of the developed scheme and its applicability to solve comple problems. AMS subject classifications: 65N6, 65N35, 76D5, 35L5 Key words: DRP scheme, compact difference scheme, computational acoustics, barrier, wave propagation. Introduction Computational acoustics problems demand space-time accurate simulation of an acoustic field over a long duration. Such problems are solved by researchers using an acoustic Corresponding author. addresses: bhumkar@iitbbs.ac.in (Y. G. Bhumkar), jp@iitbbs.ac.in (J. Pradhan),bm2@iitbbs.ac.in (B. Mahato),sakshamjindal.iitkgp@gmail.com (S. Jindal) c 2 Global-Science Press

2 2 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 analogy [] as well as by solving linearized Euler equations [2, 3]. Simulations of acoustics problems require high accuracy scheme for evaluating various derivatives present in the linearized Euler equations. Acoustic waves consist of unsteady pressure fluctuations with amplitude several order smaller compared to the background atmospheric pressure. These fluctuations display a wide band phenomenon starting from few hundred to few kilo hertz of frequencies. It is important to numerically resolve all the spatial and temporal scales in the fluid flow. Propagation of acoustic waves in air over a small distance displays non-dispersive and non-dissipative nature [4, 5]. However, most of the numerical methods are not useful for solving such problems due to their dispersive and dissipative nature. Thus, development, analysis and use of new high accuracy methods is important while solving such class of problems. Vichnevetsky [6] solved hyperbolic equations to analyze wave propagation phenomenon using finite difference approach. However, numerical dispersion analysis provided in [6] was wrong and has been later corrected in [5, 7]. Estimation of numerical phase and numerical group velocity properties to understand presence of spurious waves in the computed solution has been discussed in [7, 8]. It is not the order but the resolving ability of the numerical method that decides the accuracy of the computed solution [5, 9]. Two numerical methods can have same order of accuracy but completely different resolving abilities and the method with higher resolving ability provides more accurate results. In this regard, compact schemes are developed and widely used by different researchers for the discretization of spatial derivative terms since they offer high spectral resolution even with a relatively smaller stencil [, 26]. Ekaterinaris [] developed compact difference schemes and used them for solving hyperbolic equations corresponding to the gas dynamics and aeroacoustic test problems. Pradhan et al. [2] derived a coupled compact difference scheme(ccs) for the solution of aeroacoustic problems with an attractive feature of adaptive numerical diffusion. Resolving ability of a finite difference scheme can be further enhanced by optimization process as discussed in [3, 2 24]. For a non-periodic problem, one is forced to use different stencils at the boundary, near-boundary and the interior points. Sengupta et al. [9, 4] proposed a global spectral stability methodology to analyze central and upwind compact schemes for the estimation of the spectral resolution, numerical amplification and numerical group velocity properties of a individual grid point in a non-periodic domain. In addition to the ecellent spectral resolution for the spatial derivative terms, used numerical schemes must also ensure that the resolved scales in the computed solution propagate at the correct physical speed. Such schemes are classified as the DRP schemes [5]. The spatial and the temporal scales are linked to each other through the physical dispersion relation. Although numerical schemes solve the governing differential equations, corresponding numerical dispersion relation differs from the physical one across a complete or a band of wavenumber range due to numerical inaccuracies. The optimization of DRP schemes for solving computational acoustics problems was later followed in [2 24]. Improvements in the time integration scheme has been suggested in [5, 6] by con-

3 J. Pradhan et al. / Commun. Comput. Phys., (2), pp sidering the objective function in Eq. (.), to solve one dimensional (D) convection equation απ F= G num G eact 2 d(kh). (.) In the above equation, G num and G eact are the numerical and the eact amplification factors, respectively. Free parameter α varies between and and kh denotes nondimensional wavenumber with h as uniform grid spacing. Value of α provides limiting wavenumber up to which numerical method is epected to perform as per the eact amplification factor. Function F depends on the CFL number and coefficients of the Runge- Kutta scheme. Although these studies emphasized on use of neutrally stable methods, minimization of the phase and dispersion error was not considered in the optimization process as a constraint equation. Sengupta [5] used D wave equation as a model equation for the convection dominated flows and proposed the correct numerical dispersion relation [7] as ω N = kc N ; where ω N and C N are numerical circular frequency and numerical phase speed, respectively. Using the correct numerical dispersion relation, Sengupta et al. [8, 25, 27] devised a new strategy to spectrally optimize the coefficients of multistage time integration schemes such that the error associated with dispersion and dissipation terms was minimized for the given spatial discretization scheme. Authors in [8, 25, 27] have specifically emphasized on the fact that, the numerical properties of the DRP schemes must be evaluated by considering the spatial and the temporal discretization together. So far researchers have not optimized the spatial and the temporal discretization schemes by considering their effects together to solve D wave equation using correct numerical dispersion relation [7]. For non-periodic problems, improvements in numerical stability and DRP properties at boundary and near boundary nodes for combined compact difference (CCD) scheme have been addressed in [9, 2]. In the present work, we have first optimized the spectral resolution of the spatial discretization scheme proposed in [2]. Optimized spatial discretization scheme has been referred here as OCCS scheme. Subsequently, we have tuned coefficients of the multistage Runge-Kutta time integration scheme so that it s combination with the proposed OCCS scheme provides neutral stability even at higher CFL number along with DRP ability [8, 25, 27]. The manuscript has been organized in the following way. In the net section, detailed mathematical formulation for obtaining optimized spatial and temporal discretization scheme has been discussed. Accuracy of the proposed numerical schemes has been demonstrated by solving different computational acoustics problems in Section 3. Summary of the results and conclusions are provided in Section 4. 2 Mathematical formulation for the DRP scheme Here, authors have adopted the D wave equation as a model equation for deriving new optimized difference scheme to accurately simulate propagation of acoustic waves. Such

4 4 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 equation is always preferable due to it s non-dispersive and non-dissipative nature and is given by u u +c =, c>. (2.) t Although Eq. (2.) has non-dispersive nature, spurious waves are generated in the computed solution whenever discontinuities are present either in the solution or in the grid point distribution or in the initial and boundary conditions. These waves need to be removed from the computed solution to avoid numerical instabilities by addition of numerical diffusion [5], ( ) ( u u = 2n ) u +α d CD 2n h 2n, (2.2) where α d is a diffusion coefficient and controls amount of added numerical diffusion. Coefficient n is an integer and indicates order of added numerical diffusion. The subscript ( CD denotes central difference approimation. The term α ) 2n u d h 2n in the Eq. (2.2) is 2n responsible for addition of numerical diffusion. Solution of the Navier-Stokes equation involves estimation of the first and the second derivative terms. Authors in [2], suggested use of CCS scheme to solve aeroacoustic problems. Stencil for the CCS scheme in [2] simultaneously evaluates the first, the second and the fourth derivative terms. Addition of the fourth derivative term in an adaptive manner has been advocated in [2] in order to remove spurious unphysical waves. For a domain constructed using equi-spaced grid points with grid spacing h, the stencil in [2] is given as a (u i+ +u i )+u i +b h(u i+ u i )+c h 3 (u i+ u i ) = h d (u i+ u i ), (2.3) a 2 h (u i+ u i )+b 2(u i+ +u i )+u i +c 2h 2 (u i+ +u i ) = h 2(d 2(u i+ +u i )+e 2 u i ), (2.4) a 3 h 3(u i+ u i )+ b 3 h 2(u i+ +u i )+c 3(u i+ +u i )+u i = h 4(d 3(u i+ +u i )+e 3 u i ). (2.5) Different coefficients in Eqs. (2.3)-(2.5) are derived using the Taylor series approimations as [2] a = 3 64 ; b = 5 64 ; c = 96 ; d = ;

5 J. Pradhan et al. / Commun. Comput. Phys., (2), pp a 2 = ; b 2= ; c 2= 672 ; d 2= 24 7 ; e 2= 48 7 ; a 3 = 725 ; b 3 = ; c 3= 3 56 ; d 3= 36 7 ; e 3 = Various derivative terms in the Eqs. (2.3)-(2.5) are obtained by solving these equations in a coupled and iterative manner. For the non-periodic problems, following stencils for the boundary and the near boundary nodes have been suggested [2] u =(.5u +2u 2.5u 3 )/h, u =, u =, (2.6) u 2=(u 3 u )/(2h), u 2 =(u 2u 2 +u 3 )/h 2, u 2 =, (2.7) u 3=( u 5 +u +8(u 4 u 2 ))/(2h), u 3 =(u 2 2u 3 +u 4 )/h 2, u 3 =(u +u 5 4(u 2 +u 4 )+6u 3 )/h 4. (2.8) Finite difference schemes approimate derivatives by truncating higher order terms resulting in a difference between the eact and the numerically obtained solution due to the associated implicit filtering as shown in [5]. Any physical variable u j at the j th grid point can be epressed in the spectral plane as u j = U(k) e ik j dk. In this epression, k is a wavenumber and U(k) is the spectral amplitude associated with corresponding wavenumber. Eact derivative of function u() at the j th grid point is given as, (u j ) eact= ik U(k) e ik j dk. Numerical scheme inaccurately estimates the same derivative as(u j ) numerical= ik eq U(k)e ik j dk due to the truncation of the higher order derivative terms. Difference between the numerical and the eact value of a derivative leads to definition of the term discretization e f f ectiveness [5] which is given by the ratio of k eq /k. Real part of this effectiveness provides the spectral resolution while the imaginary part accounts for the numerical diffusion added separately by the scheme. For a numerical scheme, ideally real part of the discretization effectiveness should be equal to one across a complete wavenumber range. Different finite difference schemes, based on their stencil, deviate from this ideal value causing implicit filtering. For accurate evaluation of the spatial derivative terms, the associated implicit filtering with the CCS scheme needs to be minimized. Construction of a new high resolution OCCS scheme has been described net. 2. Optimized OCCS spatial discretization scheme Various researchers in the past [3, 2 24] derived high accuracy schemes by improving spectral resolution. In the present work, we use the same methodology for spectral optimization of the first derivative of the CCS scheme in the following way. Eq. (2.3) contains a, b, c and d as four unknown coefficients. In order to obtain these unknown coefficients, one needs four independent equations. Out of the four equations, the Taylor series epansion provides first three equations by eliminating the lead-

6 6 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 ing truncation error terms as 2a +2d =, (2.9) a +d /3 2b =, (2.) d /6 b /3 2c a /2=. (2.) However, to obtain all the four unknown coefficients in Eq. (2.3), additional algebraic equation is required. Authors in [24] have chosen this additional equation so that the difference between the actual and the numerical wavenumbers has been minimized. This new equation is combined with Eqs. (2.9)-(2.) to uniquely determine all the coefficients. Actual wavenumber for Eq. (2.3) can be given as ikh (a (e ikh +e ikh )+)+b (ikh) 2 (e ikh e ikh ) +c (ikh) 4 (e ikh e ikh ) d (e ikh e ikh ). (2.2) All the discretization schemes involve implicit filtering as the eact and numerical wavenumber differ towards the Nyquist limit. Thus, different discretization schemes can be characterized by the discretization effectiveness representation in the spectral plane. Spectral error can be minimized by matching closely the eact and the numerical wavenumbers over a complete wavenumber range [24]. One can equate the numerically obtained wavenumber k eq to those shown on the right-hand side of Eq. (2.2) ik eq h (a (e ikh +e ikh )+)+b (ik eq h) 2 (e ikh e ikh ) +c (ik eq h) 4 (e ikh e ikh )= d (e ikh e ikh ). (2.3) The quantity k eq is in general a comple quantity. Its real part is associated with the dispersion error while the imaginary part is related to the dissipation error. One can use epression for real part of k eq from Eq. (2.3) to minimize the dispersion error as shown in [24]. For achieving higher discretization effectiveness, R[k eq h] and kh must be very close across the complete wavenumber range. Authors in [2 24] constructed an error function E(kh) for optimizing resolving ability of the scheme as 7π 8 E(kh) = [( kh R[keq h] )] 2 d(kh). (2.4) For error minimization, condition E c = has been enforced. Using this constraint equation, the four unknown coefficients for the OCCS scheme are determined as a = , b = , c = , d = Fig. compares variation of discretization effectiveness R(k eq /k) with respect to the non-dimensional wavenumber kh for the OCCS scheme and the scheme in [2] for evaluating the first, the second and the fourth derivatives. Due to error minimization constraint set by Eq. (2.4), spectral resolutions for the first, the second and the fourth derivative terms show improvement with respect to the scheme in [2]. For the estimation of the

7 J. Pradhan et al. / Commun. Comput. Phys., (2), pp Real part of discretization effectiveness for the first derivative Eact Value.8 k eq /k CD2 Compact 4 CCD CCS OCCS Figure : Comparison of the variation of the real part of discretization effectiveness with respect to the nondimensional wavenumber kh for the indicated discretization schemes has been given for the first derivatives. kh first derivative, we have also shown the discretization effectiveness of the second order central difference scheme (CD2), fourth order central compact difference scheme (C4) and CCD scheme of [2] in Fig.. All these schemes including the newly derived OCCS scheme have a three point stencil. Fig. shows that OCCS scheme has highest resolution. 2.2 Optimized multistage ORK5 time integration scheme Authors in [8] have derived coefficients of multistage time integration schemes by minimizing dissipation, dispersion and phase error while solving Eq. (2.) numerically. In the present work, we have followed the same methodology for obtaining improvised time integration scheme as discussed below. Any difference scheme can be represented in the form [A]u =[B]u. This representation can be further simplified as u =[C]u where [C]=[A] [B]. Consider a computational domain divided into N equi-spaced grid points with a grid spacing h, then the first derivative is given by [5, 8], u j = h N l= C jl u l. Same epression can be represented in the spectral plane [5] as u j = h C jl U e ik( l j ) e ik j dk. (2.5) Sengupta et al. [7] defined the nodal numerical amplification factor as the ratio of spectral amplitudes of the computed solution at successive time steps as, G j = U j (k,t (n+) )/U j (k,t (n) ). For the four-stage, fourth order Runge-Kutta time integration scheme, numerical amplification factor is given as [5, 7] where A j = N c N l= C jl e ik( l j ) and N c is a CFL number. G j = A j + A2 j 2 A3 j 6 + A4 j 24, (2.6)

8 8 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 Epressions for the normalized phase velocity c N /c and normalized group velocity V gn /c are given as [5, 7] [ cn ] = β j c j ω t, (2.7) [ VgN ] = dβ j c j N c d(kh), (2.8) [ where β j = tan (G j) imag ]. The quantities(g (G j ) real j ) real and (G j ) imag indicate the real and the imaginary parts of G j, respectively. Thus the choice of spatial and temporal discretization scheme decides the important numerical properties such as numerical amplification factor along with numerical phase and numerical group velocity characteristics. Optimization of the time integration scheme has been performed by considering an autonomous system of differential equations as [8] u = f(u); t>. (2.9) t For a s-stage Runge-Kutta method, solution advancement from time level n to n+ can be given as [5, 8] u (n+) = u (n) + ( k i = t f u n + s i= i j= W i k i, a ij k j ). (2.2) The coefficients a ij s and the weights W i are chosen by first epanding terms on the left and right hand side of the Eq. (2.2) using Taylor series approimation. Subsequently multiple equations are obtained by matching the similar order terms on both sides of the equations so that the numerically obtained value u (n+) becomes close to the eact value. Rajpoot et al. [8] optimized classical eplicit Runge-Kutta methods by enhancing important numerical properties in the spectral plane and relaing the order of the scheme. In order to reduce the memory requirements during computations, we have optimized a low storage formulation of the time integration scheme given in [5, 8, 28] u (i) = u (n) +α i t f[u (i ) ], u (n+) = u (n) + t s i= W i f[u (i ) ]. (2.2) The eact amplification factor of Eq. (2.) is given as G eact = e inckh [5] while the numerical amplification factor for the Runge-Kutta method with same stages and order of accuracy is [5, 8] G num = + s j= ( ) j a j A j. (2.22)

9 J. Pradhan et al. / Commun. Comput. Phys., (2), pp The time integration scheme should be designed such that the choice of the coefficients a j s minimize the error in the wavenumber space [5, 8]. For the low storage version of RK5 method, relations between the α i s, W i s and a j s are given as W +W 2 +W 3 +W 4 +W 5 = a, (2.23) W 2 α +W 3 α 2 +W 4 α 3 +W 5 α 4 = a 2, (2.24) W 3 α α 2 +W 4 α 2 α 3 +W 5 α 3 α 4 = a 3, (2.25) W 4 α α 2 α 3 +W 5 α 2 α 3 α 4 = a 4, (2.26) W 5 α α 2 α 3 α 4 = a 5. (2.27) In the present work, we have optimized the RK5 time integration scheme ORK5 such that it s combination with the OCCS scheme provides least error while solving the D wave equation (2.). During optimization of the time integration scheme, we have fied the coefficients a = and a 2 = /2 so that conditions for the first and the second order terms in the truncation error are satisfied. The coefficients a 3, a 4 and a 5 are fied so that error in the spectral plane is minimized as described below. Following the work in [8], amplification factor for the ORK5 scheme is given as G 5 = A+ A2 2! a 3 A 3 +a 4 A 4 a 5 A 5. (2.28) Objective functions for the present OCCS ORK5 scheme are chosen such that dispersion and phase error are minimized in the spectral plane and at the same time neutral stability in the wavenumber (kh)-cfl number (N c ) plane is ensured over a larger region. Thus the important numerical properties are functions of unknown coefficients a 3, a 4 and a 5 which are optimized by setting the following constraint equations [8] 2. F (a 3,a 4,a 5,N c )= G d(kh).5, (2.29) 2. ( ) VgN F 2 (a 3,a 4,a 5,N c )= c d(kh)., (2.3) 2. ( cn ) F 3 (a 3,a 4,a 5,N c )= d(kh).. (2.3) c We have used the grid-search technique [8] to solve the constrained optimization problem. Variation of the optimized values of a 3, a 4, a 5 with respect to CFL number has been shown in Fig. 2. Although these values are function of N c, we have selected these coefficients for N c =.4, which remain more or less same for higher CFL numbers as shown in the Fig. 2. Thus in the present eercise we have chosen a 3 =.55, a 4 =.3 and a 5 =.3 as observed from the Fig. 2. One can choose different combinations of weights and αs so that Eqs. (2.23)-(2.27) are satisfied. There are total nine unknowns which include five weights and four values of α. However due to fewer available equations than the number of unknowns, we have fied five values of weights as W =W 2 =W 3 =W 4 = and

10 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 (a) a (b) Nc a (c) Nc.6.4 a Nc Figure 2: Variation of optimized coefficients a 3, a 4 and a 5 with respect to the CFL number N c has been shown in (a), (b) and (c), respectively. W 5 =. Corresponding four values of α coefficients are given by α = a 5 /a 4, α 2 = a 4 /a 3, α 3 = a 3 /a 2, α 4 = a 2 /a. Thus, the algorithm for the newly optimized low storage form of the five stage Runge-Kutta(ORK5) method is given as u () = u (n) +. t f[u () ], (2.32) u (2) = u (n) t f[u() ], (2.33) u (3) = u (n) +.3 t f[u (2) ], (2.34) u (4) = u (n) +.5 t f[u (3) ], (2.35) u (n+) = u (n) + t f[u (4) ]. (2.36) Net, we have obtained numerical properties for the OCCS ORK5 scheme for the

11 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 (a) Useful region 3 OCCS-RK4 scheme kh.5.5 Nc cr Nc (b) Useful region OCCS-ORK5 scheme kh.5..5 Nc cr Nc Figure 3: Variations of the numerical amplification factor contours for the solution of D wave equation Eq.(2.) using indicated schemes have been shown. Note that the desired neutrally stable region for the OCCS RK4 scheme is very small as compared to the OCCS ORK5 scheme. solution of Eq. (2.). Fig. 3 shows variations of the numerical amplification factor contours using indicated spatial and temporal discretization schemes. Note that the desired neutrally stable( G = ) region (as identified by a hatched region) for the OCCS ORK5 scheme is large as compared to the OCCS RK4 scheme. This is an important achievement for the newly derived optimized time integration scheme so that calculations can be performed at a much larger CFL number with retaining neutral stability. With a possibility of choosing higher CFL number for the calculations, one can use higher time step in the calculations and can perform quick computation without adversely affecting so-

12 2 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 (a) OCCS-RK4 scheme kh (b) Nc OCCS-ORK5 scheme kh Nc Figure 4: Variations of the normalized numerical group velocity contours for the solution of D wave equation Eq. (2.) using indicated spatial and temporal discretization schemes have been shown. lution quality. The critical value of the CFL number indicating the end of the neutrally stable region for the OCCS RK4 scheme is about Nc cr =.3 while the same value for the OCCS ORK5 scheme is about Nc cr =.45. This shows that with the present optimized time integration scheme, one can advance calculations more than three times faster than the traditional multistage time integration scheme. Variations of the normalized numerical group velocity contours V gn c for the solution of Eq. (2.) using indicated spatial and temporal discretization schemes have been shown in Fig. 4. The region between the contour lines.99 and. has been identified as the DRP

13 J. Pradhan et al. / Commun. Comput. Phys., (2), pp region [5]. The group velocity charts for the ORK5 scheme show a good physical dispersion relation preservation property when OCCS scheme has been used for the spatial discretization. 3 Simulations of computational acoustics problems In this section, we have performed computations of acoustic field in open and wallbounded domains using OCCS ORK5 scheme. We have first demonstrated effectiveness of the optimized OCCS ORK5 scheme over traditional difference schemes for the solution of D wave equation. Solutions of acoustic wave propagation at different instants have been reported cases by solving linearized D Euler equations. Further, we have demonstrated ability of the present discretization scheme to add a controlled amount of numerical diffusion to damp spurious oscillations as and when required while solving acoustic field set up due to monochromatic oscillations of the piston. Reflection of acoustic pulse from the solid wall in the presence and absence of rigid barrier has been addressed further to demonstrate the effectiveness of the present discretization scheme in handling comple computational acoustics problems. 3. Solution of a D wave equation Acoustic field is composed of pressure fluctuations which are usually very small as compared to the background pressure field. Propagation of acoustic signals through air shows minimal effects of viscosity [4]. Thus, the acoustic wave propagation in a D domain is non-dispersive and non-dissipative [5]. However, most of the numerical methods do not display these important properties across a complete wavenumber range as they are dispersive as well as dissipative. Calculations for computational acoustical problems usually need to be performed over a considerable duration as the information about the complete acoustic spectrum is required. The newly optimized OCCS ORK5 discretization scheme neither numerically amplify nor attenuate the acoustic signal over a considerable CFL number range as observed from the properties shown in Fig. 3. Here, we demonstrate the neutral stability and the DRP nature of the newly optimized scheme while computing at significantly higher CFL number. We have initially prescribed a wave packet centered at a significantly high wavenumber (kh = 2.2). Advantages of the newly optimized ORK5 time integration scheme over the traditional RK4 scheme are demonstrated by computing solutions at a high CFL number(n c =.4). For the present problem, we have selected a domain which has been divided by equi-spaced grid points. The initial condition used for this problem is centered at =5 and is given by u = e ( 6( 5)2) cos(kh( 5)/h), (3.) where kh=2.2 is a non-dimensional wavenumber and h is constant grid spacing.

14 4 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 (a) OCCS-RK4 scheme (b) OCCS-ORK5 scheme time =.5 time =.5 u (t) u (t) time =.5 Eact solution Numerical solution time =.5 Eact solution Numerical solution u (t) u (t) time = 2.5 Eact solution Numerical solution time = 2.5 Eact solution Numerical solution u (t) u (t) Figure 5: Solutions of D wave Eq. (2.) obtained by the indicated discretization schemes along with the eact solutions have been shown at different instants. Note that solutions obtained by using OCCS ORK5 scheme match well with the eact solution in contrast to the solutions obtained by using OCCS RK4 scheme. Solutions of Eq. (2.) are obtained by the indicated discretization schemes in Fig. 5 for the initial condition given in Eq. (3.). The eact solution at the respective instant is also shown in the corresponding frame by the dotted lines. Note that the solutions obtained using OCCS ORK5 scheme match well with the eact solution. In contrast, the solutions obtained using OCCS RK4 scheme show a large numerical error by significant attenuation of the initial condition. Thus, the advantage of the newly derived optimized spatial-temporal discretization scheme is evident as it has significantly less numerical error compared to the traditional discretization schemes. One can solve computational

15 J. Pradhan et al. / Commun. Comput. Phys., (2), pp acoustics problems for long duration quickly using OCCS ORK5 scheme by selecting large time step without adversely affecting accuracy of the computed solution. Net, we demonstrate superiority of the newly derived optimized scheme over different discretization schemes which have similar stencil size. Solutions have been obtained for the problem discussed in Fig. 5. Top frame in Fig. 6 shows the eact and the computed solutions using OCCS ORK5 scheme at t = 5 which shows an ecellent match. Bottom frames display solutions obtained at the same time using indicated difference schemes. Solutions in Fig. 6 have been obtained using same grid and at the same CFL number as in Fig. 5. It should be noted that ecept OCCS ORK5 scheme, all the difference schemes display significant numerical diffusion and dispersion error. Fig. 6 also shows that irrespective of use of either RK4 or ORK5 time integration scheme, solutions obtained using CD2, C4 and CCD compact schemes display large numerical error. Only combination of OCCS ORK5 scheme displays appreciable results. This aspect clearly shows that one needs to optimize spatial and temporal discretization schemes together as eplained in the present manuscript as compared to the traditional optimization approaches available in the literature. 3.2 Reflection of pressure waveform Here, we consider propagation of a acoustic disturbance following D linearized Euler equations in a dimensional form. Present problem helps to demonstrate the effectiveness of numerical scheme while simulating acoustic wave reflection cases. The governing equations in the Cartesian coordinates are given as u t + p =, ρ (3.2) p t +γp u =. (3.3) We have considered a domain 7.5m 7.5m and divided it into 5 equi-spaced grid points. Time step for the present problem has been chosen as t=.s. Solutions for Eqs. (3.2)-(3.3) have been obtained using OCCS ORK5 scheme. For the present problem, the initial condition is prescribed as ( ) 4π u = ; p = e 52 sin. (3.4) 3 The mean atmospheric pressure P and mean atmospheric density ρ have been prescribed as 325 N/m 2 and.225 kg/m 3, respectively. The ratio of the specific heats has been indicated by γ=.4. Fig. 7 shows propagation of an acoustic disturbance following the Eqs. (3.2)-(3.3) at the indicated instants. In this case, solutions for the Eqs. (3.2)-(3.3) have been obtained at all the grid points at every time step with prescription of reflecting boundary condition( p = ) at the boundaries. Top frame of Fig. 7 shows the initial condition. The acoustic disturbance splits in to two waveforms each traveling in opposite

16 6 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 OCCS-ORK5.5 time = 5 Numerical Solution Eact Solution u(t) CD2-RK4.5 time = 5 Numerical Solution Eact Solution CD2-ORK5.5 time = 5 Numerical Solution Eact Solution u(t).5 u(t) C4-RK4.5 time = 5 Numerical Solution Eact Solution C4-ORK5.5 time = 5 Numerical Solution Eact Solution u(t).5 u(t) CCD-RK4.5 time = 5 Numerical Solution Eact Solution CCD-ORK5.5 time = 5 Numerical Solution Eact Solution u(t).5 u(t) Figure 6: Solutions of D wave Eq. (2.) obtained by the indicated discretization schemes along with the eact solutions have been shown at time = 5. Note that, solutions obtained by OCCS ORK5 scheme match well with the eact solution in contrast to the solutions obtained by other spatial difference schemes with similar stencil size. direction. The disturbance gets reflected from the domain boundaries and recombine as shown in the bottom most frame. There is an ecellent match between the analytical and numerical results as shown at different instants.

17 J. Pradhan et al. / Commun. Comput. Phys., (2), pp time = S Eact solution Computed solution.5 P (Pa) (m) 5 time =. S Eact solution Computed solution.5 P (Pa) (m) 5 time =.2 S Eact solution Computed solution.5 P (Pa) (m) 5 time =.3 S Eact solution Computed solution.5 P (Pa) (m) 5 time =.44 S Eact solution Computed solution.5 P (Pa) (m) 5 Figure 7: Comparison of the analytical and numerical solutions of Eqs. (3.2)-(3.4) obtained using OCCS ORK5 scheme at the indicated instants. Comparison shows an ecellent match between the analytical and numerical results and shows the applicability of the proposed scheme to compute computational acoustics problems accurately.

18 8 J. Pradhan et al. / Commun. Comput. Phys., (2), pp Sound radiation by an oscillating circular piston In this problem, monochromatic oscillations of a circular piston result in the formation of acoustic field. We have considered a domain with 4 and 2 r 2 with 4 grid points in each direction. The wall is located at = while the piston has been kept in between r = and r =. Thus, the radius of the piston is and the boundary conditions on the wall surface = are given as [29] u=, r >, ( ) πt u=.sin, r. 5 The linearized ai-symmetric Euler equations in the cylindrical coordinates (r, ) for this model problem are given as [29] u t + p =, (3.5) v t + p =, r (3.6) p t + v r + v r + u =. (3.7) Optimized OCCS ORK5 scheme has been used to evaluate the derivatives in Eqs. (3.5)-(3.7). Computations are performed with a time step of t =.. As the piston moves forward along the X-ais, fluid immediately net to the face of the piston gets ecitation and moves along with the piston. Pressure in the fluid element very close to piston surface increases. This element epands in the forward direction displacing the net layer of fluid and causing its compression. Thus a pressure pulse is formed which travels at the speed of sound. As the piston oscillates, acoustic field composed of compression and rarefaction zones is formed. Monochromatic oscillations of the piston, set up an acoustic field in the domain as shown in Figs. 8(a)-(b) at the indicated instants. Acoustic disturbances travel radially outwards and the acoustic field displays symmetry about the ais r=. Due to discontinuity in the boundary condition at the boundary =, spurious waves setup in the domain. These waves remain present in the solution as shown in Fig. 8(a), when one does not add any numerical diffusion. However, if we add the fourth order diffusion term to the solution, spurious unphysical waves which are essentially made up of high wavenumber components are eliminated as shown in Fig. 8(b). Computed pressure fluctuations on the piston centerline r= have been compared with the numerical results in [29] at the indicted instant. Here, an ecellent match in the comparison between present computed solution and Fung s [29] result have been obtained and shown in Fig. 8(c). Thus the newly derived optimized scheme is useful for solving the computational acoustic problems.

19 J. Pradhan et al. / Commun. Comput. Phys., (2), pp (a) Calculations without numerical diffusion coefficient t = 4 t = 7 Spurious q-waves t = Spurious q-waves 5 Spurious q-waves 5 5 r r r (b) Calculations with numerical diffusion coefficient. t = 4 t = 7 t = r r r (c).2 t = 9 Present results Results of Fung et. al p Figure 8: Solutions for the case of the acoustic field radiated by an oscillating circular piston have been shown here. Frames (a) and (b) show pressure contours for the calculations performed without and with adding numerical diffusion, respectively. Variation of the acoustic pressure on line r = has been compared with the results of [29], in (c). Note that the unphysical spurious waves have been eliminated in frame (b) by adding numerical diffusion. 3.4 Reflection of an acoustic pulse over a solid wall Here, we have considered a case of development of acoustic field in the presence of mean flow in a semi-infinite space. For the present problem, we have selected a two dimen-

20 2 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 sional computational domain with, y 2. The wall has been located at y=. An acoustic pulse displays reflection due to the presence of a solid wall. The Euler equations in non-dimensional form are given by [29] ρ t + (Mρ+u) + v =, y (3.8) u (Mu+ p) + =, t (3.9) v t + (Mv) + p =, y (3.) p t + (Mp+u) + v =, y (3.) where M=.5 denotes Mach number. The initial condition for the present problem is given as u = v =, p = ρ = ep ( (ln2) 2 +(y 25) 2 25 ). (3.2) This initial condition has been shown in the top frame of Fig. 9. Here, we have divided domain with 2X2 equi-spaced grid points and computations are performed using a time step t=.5. On the bottom surface of the domain(y=), we have applied a reflecting boundary condition. Computations are also performed by adding numerical diffusion with diffusion coefficient. to attenuate any spurious high wavenumber components. Time evolution of the initial disturbance under the action of mean flow has been shown in the left side frames obtained by using the present optimized schemes. Present results are compared with the results in [29] at the identified instants. Comparison shows ecellent match. Thus the present scheme successfully computes acoustic wave reflection for a problem which also involves a mean flow. 3.5 Effects of rigid barrier on the development of an acoustic field Net, we have studied development of an acoustic field in the presence and absence of the rigid barrier. Analysis of the acoustic field in the presence of barrier has been objective of large number of studies [32 37]. Noise control using barriers of different materials, shapes and sizes is an important research topic even today. For such problems, physical phenomena such as wave reflection, diffraction occur whose numerical prediction poses a challenge. In this case, solutions of acoustic field triggered by harmonic ecitation in the presence and absence of barrier have been computed. Propagation of small acoustic disturbances is governed by linearized Euler equations. These equations can be written

21 J. Pradhan et al. / Commun. Comput. Phys., (2), pp (a) 2 5 Level PRE 2.5. OCCS-ORK5 t=3 Y X (b) 2 5 Level PRE 2.5. OCCS-ORK5 t= (c) t= Y 5 Fung et. al X (d)..5 OCCS-ORK5 t=6 (e) P X (f) OCCS-ORK5 t=.5 P -2 P -.5 Fung et. al. 2 4 X 6 8 Figure 9: Reflection of an acoustic disturbance from a flat plate located at y = following non-dimensionalized 2D Euler Eqs. (3.8)-(3.) has been shown. Pressure contours with levels. and.5 have been compared with the results in [29] at the identified instants. Comparison shows ecellent match.

22 22 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 in a non-dimensional form as [3] ρ t + u + v y = S, (3.3) u t + p =, (3.4) v t + p =, y (3.5) p t + u + v y = S. (3.6) Above equations are non-dimensionalized by taking as a reference length scale, ambient sound speed c as a reference velocity scale, /c as the time scale, ρ as a density scale, and ρ c 2 as the pressure scale. The source term S is given by [ S = ep ln2 ( ( ) 2 +(y y ) 2 r 2 )] cos(ωt), (3.7) where =y =.5 and ω=π. For the present problem, we have chosen a two dimensional domain , y 4. Grid points in the domain are equi-spaced. There are 9 4 grid points with a chosen time step of t=.5. We have prescribed a solid wall at y = by giving a reflecting boundary condition. Radiation boundary condition has been prescribed on the remaining three sides of the domain. In this study, we have considered development of an acoustic field in the absence and presence of rigid barrier (centered at =). For a case with an acoustic barrier, thickness of the barrier is chosen as.4 while the height of the barrier is.. Although the choice of source resembles monopole which does not introduce directionality in the computed solution, prescription of reflective boundary condition at the bottom wall (y=) and the barrier surface induces directionality in the acoustic field. Fig. shows contours of disturbance pressure at time t=9.8 for the case without and with barrier in (a) and (b), respectively. For the case without barrier, one observes symmetric acoustic field across line =.5. This symmetry is absent for the case with barrier. Comparison of the time histories of acoustic pressure variations at the identified points on both sides of the barrier with that of no barrier case has been shown in (c) and (d). Time history at location = and y=.5 situated on the left side of the barrier, shows that, case with barrier displays higher pressure amplitude variation as compared to the without barrier case. In contrast, the time histories for location = 2. and y =.5 shows that the pressure amplitude variation for the case with barrier has almost three times smaller amplitude as compared to that of without barrier case. Fig. (a) and (b) show contours of RMS value of pressure for the case without and with barrier, respectively. One can clearly observe the shadow region behind the barrier. For the case without barrier, contours for RMS value of pressure display symmetry along line =.5. Variation of RMS value of pressure at unit radial distance from the eciter

23 J. Pradhan et al. / Commun. Comput. Phys., (2), pp Figure : Solutions of the non-dimensionalized, 2D Euler Eqs. (3.3)-(3.6) have been shown here. Contours of disturbance pressure at time t=9.8 have been compared for the case without and with barrier in (a) and (b), respectively. For the case without barrier, one observes symmetric acoustic field across line =.5. This symmetry is absent for the case with barrier. Comparison of the time histories of acoustic pressure variations at the identified points on both sides of the barrier with that of no barrier case has been shown in (c) and (d). location =y=.5 with an angle θ has been shown in Fig. (c). For the case without barrier, a main lobe at angle θ= 9 o has been observed. In addition, two side lobes are also observed on either side of the main lobe for the case without barrier. In contrast for the case with barrier, width of a main lobe is significantly reduced and maimum RMS

24 24 J. Pradhan et al. / Commun. Comput. Phys., (2), pp (a) y (b) y (c) 2 9 Without barrier case 2 With barrier case (theta) 8.. p.2 RMS Figure : Solutions of the non-dimensionalized, 2D Euler Eqs. (3.3)-(3.6) have been shown here. Contours of RMS value of pressure have been shown for the case without and with barrier in (a) and (b), respectively. One can clearly observe the shadow region behind the barrier as compared to the region ahead of the barrier. Variation of RMS value of pressure at unit radial distance from the eciter location = y =.5 for the case with and without barrier has been shown in (c). value of the pressure associated with the main lobe is significantly higher as compared to the case without barrier.

25 J. Pradhan et al. / Commun. Comput. Phys., (2), pp Summary and conclusions Here, we have proposed a new high accuracy DRP scheme to solve computational acoustics problem accurately. Optimization of the scheme has been carried out by considering space and time discretization together. The derived scheme has following advantages: Near spectral resolution helps to compute with fewer grid points over a considerable wavenumber range. For the developed scheme, neutral stability is observed up to Nc cr =.45 which allows to perform computations at significantly higher time step as compared to the traditional discretization schemes. Scheme displays DRP behavior over a considerable wavenumber range. The ORK5 time integration scheme is based on low storage formulation and requires less memory. The method has ability to add numerical diffusion in an adaptive manner to remove numerical instabilities. We have performed accurate computations of sound in open and wall-bounded domains using present OCCS ORK5 scheme as shown in Figs. 7-. In these eamples, we have demonstrated ability of the present discretization scheme to add a controlled amount of numerical diffusion to damp spurious oscillations as and when required. Cases involving reflection of acoustic pulse from the solid wall in the absence and presence of barrier have been also discussed to demonstrate effectiveness of the present discretization scheme in handling comple computational acoustics problems. Acknowledgments We acknowledge the Department of Science and Technology, India for providing support through the grant SB/FTP/ETA-2/24 to carry out the present work. References [] X. Jiang, E. J. Avital, and K. H. Luo, Direct computation and aeroacoustic modelling of a subsonic aisymmetric jet, Journal of Sound and Vibration, 27 (24), [2] C. Bailly, and D. Juve, Numerical solution of acoustic propagation problems using linearized Euler equations, AIAA Journal, 38 (2), [3] E. J. Avital, N. D. Sandham, and K. H. Luo. Mach wave radiation by miing layers. Part I: Analysis of the sound field, Theoretical and Computational Fluid Dynamics, 2, no. 2 (998), [4] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics, ISBN Wiley-VCH, December (999), 56.

26 26 J. Pradhan et al. / Commun. Comput. Phys., (2), pp. -27 [5] T. K. Sengupta, High Accuracy Computing Methods: Fluid Flows and Wave Phenomena, Cambridge University Press, (23). [6] R. Vichnevetsky, Wave propagation analysis of difference schemes for hyperbolic equations: a review, International Journal for Numerical Methods in Fluids, 7, no. 5 (987), [7] T. Poinsot, and D. Veynante, Theoretical and Numerical Combustion, RT Edwards, Inc., (25). [8] T. K. Sengupta, Y. G. Bhumkar, M. Rajpoot, V. K. Suman and S. Saurabh, Spurious waves in discrete computation of wave phenomena and flow problems, Applied Mathematics and Computation, 28 (22), [9] T. K. Sengupta, S. K. Sircar, and A. Dipankar, High accuracy schemes for DNS and acoustics, Journal of Scientific Computing, 26, no. 2 (26), [] S. K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 3, no. (992), [] J. A. Ekaterinaris, Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics, Journal of Computational Physics, 56, no. 2 (999), [2] J. Pradhan, B. Mahato, S. D. Dhandole, and Y. G. Bhumkar, Construction, analysis and application of coupled compact difference scheme in computational acoustics and fluid flow problems, Communications in Computational Physics, 8, no. 4 (25), [3] G. Ashcroft, and X. Zhang, Optimized prefactored compact schemes, Journal of Computational Physics, 9, no. 2 (23), [4] T. K. Sengupta, G. Ganeriwal, and S. De. Analysis of central and upwind compact schemes, Journal of Computational Physics, 92, no. 2 (23), [5] M. Bernardini, and S. Pirozzoli, A general strategy for the optimization of RungeKutta schemes for wave propagation phenomena, Journal of Computational Physics, 228, no. (29), [6] S. Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave propagation problems, Journal of Computational Physics, 222, no. 2 (27), [7] T. K. Sengupta, A. Dipankar, and P. Sagaut, Error dynamics: beyond von Neumann analysis, Journal of Computational Physics, 226, no. 2 (27), [8] M. K. Rajpoot, T. K. Sengupta, and P. K. Dutt, Optimal time advancing dispersion relation preserving schemes, Journal of Computational Physics, 229, no. (2), [9] T. K. Sengupta, V. Lakshmanan, and V. V. S. N. Vijay, A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, Journal of Computational Physics, 228, no. 8 (29), [2] T. K. Sengupta, V. V. S. N. Vijay, and S. Bhaumik, Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties, Journal of Computational Physics, 228, no.7 (29), [2] C. H. Yu, Y. G. Bhumkar, and T. W. H. Sheu, Dispersion relation preserving combined compact difference schemes for flow problems, Journal of Scientific Computing, 62, no.2 (25), [22] Y. G. Bhumkar, T. W. H. Sheu, and T. K. Sengupta, A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations, Journal of Computational Physics, 278 (24), [23] T. W. H. Sheu, Y. G. Bhumkar, S. T. Yuan, and S. C. Syue, Development of a High-Resolution Scheme for Solving the PNP-NS Equations in Curved Channels, Communications in Computational Physics, 9, no.2 (26), [24] P. H. Chiu, and T. W. H. Sheu, On the development of a dispersion-relation-preserving

27 J. Pradhan et al. / Commun. Comput. Phys., (2), pp dual-compact upwind scheme for convectiondiffusion equation, Journal of Computational Physics, 228, no. (29), [25] T. K. Sengupta, M. K. Rajpoot, and Y. G. Bhumkar, Space-time discretizing optimal DRP schemes for flow and wave propagation problems, Computers & Fluids, 47, no. (2), [26] R. Hion, and E. Turkel, Compact implicit MacCormack-type schemes with high accuracy, Journal of Computational Physics, 58, no. (2), 5-7. [27] R. Bose, T. K. Sengupta, Analysis and design of a new dispersion relation preserving alternate direction bidiagonal compact scheme, Journal of Scientific Computing, 64, no. (25), [28] D. Stanescu, and W. G. Habashi, 2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 43, no. 2 (998), [29] K. Y. Fung, S. O. M. Raymond, and S. Davis, A compact solution to computational acoustics, In ICASE/LaRC workshop on benchmark problems in computational aeroacoustics (CAA), (995). [3] C. W. Lim, C. Cheong, S-R. Shin, and S Lee. Time-domain numerical computation of noise reduction by diffraction and finite impedance of barriers, Journal of Sound and Vibration, 268, no. 2 (23), [3] C. Lee, and Y. Seo, A new compact spectral scheme for turbulence simulations, Journal of Computational Physics, 83, no. 2 (22), [32] Z. Maekawa, Noise reduction by screens, Applied Acoustics,, no. 3 (968), [33] U. J. Kurze, Noise reduction by barriers, The Journal of the Acoustical Society of America, 55, no. 3 (974), [34] H. G. Jonasson, Diffraction by wedges of finite acoustic impedance with applications to depressed roads, Journal of Sound and Vibration, 25, no. 4 (972), [35] D. C. Hothersall, S. N. Chandler-Wilde, and M. N. Hajmirzae, Efficiency of single noise barriers, Journal of Sound and Vibration, 46, no. 2 (99), [36] R. Seznec, Diffraction of sound around barriers: use of the boundary elements technique, Journal of Sound and Vibration, 73, no. 2 (98), [37] D. Duhamel, Efficient calculation of the three-dimensional sound pressure field around a noise barrier, Journal of Sound and Vibration, 97, no. 5 (996),