IMPICE Method for Compressible Flow Problems in Uintah.

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids ; : 44 Published online in Wiley InterScience ( DOI:./fld Method for Compressible Flow Problems in Uintah. L.T. Tran,,, and M. Berzins,, School of Computing, University of Utah, Salt Lake City, UT 84, U.S.A. The Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84, U.S.A. SUMMARY The Implicit Continuous-fluid Eulerian (ICE method is a successful and widely used semi-implicit finitevolume method that applies to flows that range from supersonic to subsonic regimes. The classical ICE method has been epanded to problems in multiphase flow which span a wide area of science and engineering. The ICE method is utilized by the C-SAFE code Uintah written at the University of Utah to simulate eplosions, fires and other fluid and fluid-structure interaction phenomena. The ICE method used in Uintah (referred to here as Production ICE is described in many papers by Kashiwa at Los Alamos and Harman at Utah. However, Production ICE does not perform as well as many current methods for compressible flow problems governed by the Euler equations. We show, via eamples, that changing the nonconservation form of the solver in Production ICE to a conservation form improves the numerical solutions. In addition, the use of slope limiters makes it possible to suppress the nonphysical oscillations generated by the ICE method in conservation form. This new form of ICE is referred to as, the IMproved Production ICE method. The accuracy of for one dimensional Euler equations is investigated by using a number of test cases. Copyright c John Wiley & Sons, Ltd. Received... KEY WORDS: Implicit Continuous-fluid Eulerian (ICE method; Numerical method for Compressible Flow Problems; Improved ICE method using limiters; Method for Compressible Flows; Improve Production ICE in Uintah; spatial and temporal errors.. INTRODUCTION The ICE method was developed by Harlow and Amsden in 968 [] with the aim of calculating the flows in all velocity ranges. The key idea for the method is to use a semi-implicit time discretization, in which the acoustic waves are treated implicitly while the advection terms are treated eplicitly. As a result, the ICE method is able to remove the Courant stability limitation based on the speed of sound in the fluid. According to Harlow and Amsden [], this is a numerically stable and efficient method for calculating transient, viscous fluid flows in several space dimensions. Harlow ltran@cs.utah.edu mb@cs.utah.edu Correspondence to: L.T. Tran, School of Computing, University of Utah, 5 S. Central Campus Drive, Salt Lake City, UT 84, U.S.A. Copyright c John Wiley & Sons, Ltd. [Version: /5/3 v.]

2 L.T. TRAN, M. BERZINS and Amsden in 97 [] simplified the method and also greatly etended its scope of applicability. The ICE method by Harlow and Amsden was intended to simulate single-phase fluid dynamics problems. There are several improved versions of the ICE method using a pressure-correction solution procedure as seen in [4, 8, 9, 3, 37], and one typical pressure-correction method is referred as PISO (Pressure Implicit with Splitting of Operations. The ICE method was later etended by Harlow and Amsden in 975 [3] and Kashiwa et al. in 994 [] to work with multiphase flow simulations. The ICE method by Harlow and Amsden used a staggered grid with normal velocity components at cell faces and all other variables at cell centers. As mentioned in Kashiwa and Lee [], the main difficulties in the use of the staggered mesh include adding the artificial terms corresponding to a bulk viscosity to the equations in order to obtain reasonably smooth variation in density near shock waves and the development of spurious fluid as a result of a purely nonphysical circumstance. Kashiwa et al. [] devised a new implementation of the ICE method by using a cell-centered scheme in which velocity is also located at the cell-center in an ongoing effort to deal with the difficulties in the classical approach. In the cell-centered approach, the velocity at cell faces is not computed directly, but is defined using the flow field or other dependent variables. Kashiwa and Lee [] mention that definition of the face-centered velocity in the cell-centered scheme is a crucial matter for the robustness of the method. The fully cell-centered implementation of the ICE method in Kashiwa et al. [] employs a conservative advection operator and a Lagrangian part which leaves a degree of freedom in the choice of conservation variables. The conservation laws used include at least those of mass, linear momentum, and internal energy (or alternatively the total energy. The Lagrangian part in most standard ICE implementations is fully conservative and it usually conserves the internal energy rather than the total energy. Due to its general applicability, the ICE method has been used to study numerous comple flow problems, for eample, fluidized dust beds, the flow of a liquid with entrained bubbles, atmospheric condensation with the fall of precipitation, the epansion and compression of a bubble formed by high-eplosive gases under water and dynamics resulting from intense atmospheric eplosions from the early time highly compressible flow [3, ]. With its ability to handle comple flow problems, the ICE method for multiphase flows is also utilized by the C-SAFE code Uintah written at the University of Utah to simulate eplosions, fires and other fluid and fluid-structure interaction phenomena [9]. The ICE code in the Uintah Framework will be referred to hereafter as Production ICE. The implementation of Production ICE is based on the cell-centered ICE method by Kashiwa et al. [] with a few eceptions that will be discussed in detail in Section 3. The numerical scheme used in Production ICE [9,, 4, 7, 8] solves the conservation of mass, linear momentum and internal energy. However, the Lagrangian part in Production ICE is in non-conservative form which appears to be an eception to the standard ICE method. While this may not be a problem for some cases, it appears to be a problem when applying this Production ICE code to single-fluid cases that are governed by the Euler equations in which the obtained numerical solutions ehibit some discrepancies in the shock speeds and they additionally show unphysical oscillations. The Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

3 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 3 Production ICE method for single-fluid cases solves the fluid flows that are governed by the onedimensional, time-dependent Euler equations of gas dynamics given by ρ ρ ρu + u ρu ρe ρe t + p =, ( pu where ρ(,t is the density, u(,t is the velocity, p(,t is the pressure, and e(,t is the internal energy per unit mass. The following equation of state is also employed: where γ is the specific heat ratio with the value of.4 for ideal gas. p = (γ ρe, ( It has been mentioned in [7, 6, 35] that nonconservative schemes approimating hyperbolic conservation laws do not converge to the correct solution in general. So the eistence of discrepancies in the numerical solutions for non-linear hyperbolic systems using Production ICE is quite understandable. Therefore, in order to improve the Production ICE method, we first change the method to solve the system of one-dimensional Euler equations in conservation form where the total energy instead of the internal energy is conserved. The Improved Production ICE method, will be referred to hereafter as, is a cell-centered ICE method which solves the system of one-dimensional Euler equations in conservation form that is given by and the equation of state is given by ρ ρ ρu + u ρu + p ρe ρe pu t =, (3 p = (γ (ρe ρu, (4 where E(,t is the total energy per unit mass and E = e+ u. As a result of changing the Production ICE method to solve the system of Euler equations in conservation form, the computational results show the disappearance of the discrepancies in the obtained numerical solutions (refer to Appendi C. In Appendi C, the original method of Kashiwa et al [] is referred to as Cell-centered ICE. However, Cell-centered ICE suffers from unphysical oscillations when there are moving contact discontinuities. Typically, methods in the literature use a variety of techniques such as constrained data reconstruction so as to avoid spurious oscillations; for eample, [,, 5, 39, 4, 4, 4]. To suppress oscillations, we will use a similar approach in which the data at the cell interface is the approimate Riemann solution to the local Riemann problem that is constructed by using slope-limited interpolation of the left and right cell-centered data. The approimate Riemann solver which was proposed by Harten et al. [5] and used by Davis [6] to satisfy consistency with the integral forms of the conservation law and entropy condition will be used to solve the local Riemann problem. The slope limiter used is selected from the etensive literature on the functions for slope limiters in the last few decades; see, for eample, Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

4 4 L.T. TRAN, M. BERZINS [7, 6, 33, 36, 39, 4, 4, 43]. The method is Cell-centered ICE with the oscillations being suppressed using the above-mentioned technique. In effect, although the original ICE method is a von-neumann type method in which the flues are fully dependent on the time increment, we have now introduced a Riemann solver approach in which the flues depend directly on the approimate solution to the Riemann problem. The discussion of the methods in this paper will need the definition of the speed of sound. The speed of sound, c(,t, in an ideal gas is defined by c = γp ρ. (5 As for many numerical methods for the solution of PDEs, the method approimates the Euler equations by a finite volume method using a spatial discretization of the problem and a time integration technique. Besides the importance of having a non-oscillating numerical solution, the numerical accuracy of the method in time and space is also important. The method is first order in space and time. However, first-order methods are known to be not accurate enough to be used for large problems on relatively coarse grids. In order to increase the order of accuracy in time and space, we need to employ a nonlinear spatial discretization combined with a high order time discretization. Our goal is to obtain an method with second-order accuracies in both time and space. In this paper, we will study numerical accuracy of the method and discuss how to change the method to achieve second-order accuracy in both space and time. The content of this paper is organized as follows. In Section, we recap the cell-centered ICE method by Kashiwa et al. [] which includes the spatial discretization and essential steps in the time integration. In Section 3, we present a detail implementation of the Production ICE method and describe differences between it and the cell-centered ICE method by Kashiwa et al. []. In Section 4, we discuss the proposed method by Kwatra et al. [3] that can be applied to calculate the time integration step of the semi-implicit ICE method. This method removes the restriction of sound speed in calculating the time step, but still maintains stability. In Section 5, we propose a modification to the Production ICE method to remove the unphysical oscillations in the numerical solutions. The numerical solutions of the method are presented and compared to the numerical solutions of the Production ICE method in Section 6. The spatial and temporal accuracies of the method are shown in Section 7. The issue of how to obtain second-order accuracy in time and space is discussed in Section 8 and Section 9. As it will be discussed in Section, there are many degrees of freedom in the implementation of the cell-centered ICE method by Kashiwa et al [], so the purpose of this paper is not only to find an improved implementation of the Production ICE method but also to discuss how various choices in the implementation of the cell-centered ICE method affect the obtained numerical solutions.. THE CELL-CENTERED ICE METHOD BY KASHIWA ET AL. [].. General description of the cell-centered ICE method The cell-centered ICE method described in detail in [] is a finite-volume solver in which space is discretized into N uniform cells of width = (b a/n where [a,b] is the spatial domain. The Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

5 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 5 cells are centered around = a+( / where =..N. The boundaries of these cells are located at + = a+ where =..N and also called face-centers or cell interfaces. With this discretization, the domain boundaries are aligned with the first and last cell edges. A time integration method is used to estimate the averages of cell variables at some time t = T end from the averages of cell variables at t =. For each time integration step, assuming that the cell averages at time t n are known, the goal is to compute the averages of cell variables at the net time step t n+. The ICE method invokes operator splitting in which the solution consists of a Lagrangian phase and an Eulerian phase. The Lagrangian phase advances cell values without advection and maps new values to cell variables and the Eulerian phase advects the cell variables. The essential point that makes the ICE method an all-speed scheme is to use an implicit scheme for the Lagrangian phase and an eplicit scheme for the Eulerian phase. The cell-centered ICE method comprises the following phases: The Primary Phase: With the spatial discretization as discussed above, the spatial derivatives in the governed equations are approimated using finite differences of quantities at face-centers. Since an implicit scheme is used for the Lagrangian phase, the variables involved in the Lagrangian phase are those evaluated at face-centers att n + t and are determined in this phase. It is also necessary in this phase to estimate the fluing velocity which is going to be used in the Eulerian phase. The fluing velocity, u, is + the flu of volume across the cell interface. In order to make clear which variables are defined at face center, the superscript is used here for these variables as required. The Lagrangian Phase: Let V n be the volume of cell and U n be the vector of averaging cell variables at t n. In particular, V n is equal to for the above discretization. Assume that the cell volume is changed during the Lagrangian phase to V L and V L = V n + t(u u where u and u are fluing + + velocities at cell interfaces. There is also a change in the vector of averaging cell variables to U L after the Lagrangian phase has been completed. A numerical scheme obtained from neglecting the convective effects is used to evaluate the change in the material state and in turn evaluate U L. The Eulerian Phase: For this phase, we have to evaluate the change in the solution due to advection. LetV n+ be the cell volume att n+ and assume that the mesh is stationary, thenv n+ =. The change in the solution due to advection is as follows V n+ U n+ = V L U L t(u + U + u U, (6 where U + = tn+ t t n U( +,tdt is the vector of advected quantities and is numerically determined. As suggested by [], this numerical value may be determined using U n or U L; however, how to numerically determine this is not eplicitly presented. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

6 6 L.T. TRAN, M. BERZINS State Variables Update Phase: In this phase we update the primitive cell variables using U n+ will be used for the following time integration step. and the equation of state as these.. The cell-centered ICE implementation of Kashiwa et al. [] Kashiwa et al. s improvements, [], to ICE consist of changes to the above primary and Lagrangian phases. There are two important quantities that will be used in these phases, these are the facecentered velocity and pressure as denoted by u and p. The face-centered velocity, u, is also the advected speed for the Eulerian phase, so [] refers this as the fluing velociy. The facecentered fluing velocity, u, is calculated based on the time-advanced equation for velocity. For + the system of the Euler equations, the derived equation for velocity evolution is Equation (7 is written in Lagrangian form as u t +uu = p ρ. (7 Du Dt = p ρ. (8 The fluing velocity in the cell-centered ICE method is obtained using the flow field and semiimplicit Euler scheme in the Lagrangian frame by u + = u n + ρ t p n+ ρ n + + pn+, (9 where p n+ is the cell-centered pressure at t n + t, un + ρ is the mass-weighted average facecentered velocity and ρ n is the average face-centered density at time t + n. The mass-weighted average velocity, u n + ρ, of left and right states at face-center is given by u n + ρ = ρn un +ρn + un + ρ n, ( +ρn + and the average face-centered density, ρ n +, of left and right cell-centered densities is: As the face-centered pressure att n + t ρ n +, pn+ = ρn +ρn +. (, is not readily available, it needs to be obtained from correcting the face-centered pressure att n, p n. Letδpn = pn+ p n be the difference between the cell-centered pressures at these two time levels and which is referred to in Kashiwa et al. [] as pressure corrector, equation (9 is then rewritten as u + = ũ + t δp n + δpn ρ n, ( + Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

7 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 7 where ũ + = u n + ρ t p n + pn ρ n. (3 + In order to determine the face-centered fluing velocity,u, the pressure corrector values,δp n + + and δp n, need to be determined using the equation of pressure evolution, which is written in Lagrangian form as p t +up = c ρu, (4 Dp Dt = c ρu. (5 In [], two different ways to determine the pressure corrector values are discussed. The eplicit pressure corrector uses the eplicit discrete form and the implicit pressure corrector uses the semi-implicit discrete form of (5. The eplicit form of pressure corrector is given by δp n = t un ( p n + p n The implicit pressure corrector is obtained using δp n = t un ( p n + p n t (c ρ n ( u n + ρ u n ρ. (6 t (c ρ n (u + u (7. The implicit pressure corrector is complicated since equations ( and (7 show that the facecentered fluing velocities and the pressure corrector values are interrelated, so there is a need to calculate the fluing velocities using these two equations and this is not eplicitly discussed in []. After having determined the pressure corrector values at cell centers, the face-centered fluing velocity, u, is determined using equation ( where ũ is defined in ( There are several suggested choices for calculating the face-centered pressure, p, in [] and + []. Two of these choices, which are derived from the pressure equation in Kashiwa et al in 994 [], will be discussed in Appendi D. In this paper, we employ the choice that is mentioned in Kashiwa in []. This choice aims to satisfy continuity of acceleration by equating acceleration increments for the left/right half spaces. The equation as specified in [] is: p + = ρ n + p n+ + + ρ p n+ n ρ n + ρ n +. (8 In the above equation, the face-centered pressure is thus calculated using specific volumes-weighted of the left and right time-advanced cell-centered pressures. The Primary Phase is eecuted after the face-centered velocity and pressure, u and p, + + are calculated. The choice of the vector of conserved variables, U, and the numerical procedure to determine the vector of the face-centered advected quantities, U +, are neccessary for the implementation of the Lagrangian phase and the Eulerian phase. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

8 8 L.T. TRAN, M. BERZINS 3. PRODUCTION ICE IN UINTAH COMPUTATIONAL FRAMEWORK The term Production ICE is used to denote the ICE method as implemented in Uintah Computational Framework by [9,, 4, 7, 8] to simulate fluid flows that are governed by the Euler and Navier Stokes equations. Production ICE solves the Euler system in non-conservation form ( with the vector of variables U = [ρ,ρu,ρe] T. The detail implementation of the phases in Production ICE follows the description given in Kashiwa et al. [] with some eceptions that will be pointed out eplicitly in the following discussion. 3.. The Primary Phase The first eception is that the face-centered quantities in Production ICE are not time-centered. The face-centered fluing velocity, u, and pressure, p, for the time step [t + + n,t n+ ] are approimated at the face-center at time t n+. For this reason, we use the notations u and p + + for the face-centered fluing velocity and pressure in Production ICE. Using a different approach from equation (9, the face-centered fluing velocity in Production ICE, u, is approimated as: + u + = u n + ρ t p n + pn ρ n, (9 + where the mass-weighted average velocity and the average face-centered density, u n + ρ and ρ n +, are defined in ( and (. So another eception in calculating the fluing velocity in Production ICE is that the scheme in (9 is not semi-implicit when the pressures used are defined at t n. The face-centered pressure, p +/, in Production ICE is calculated using the following equation p + = ( ρ n + p n+ + + ρ n ρ n + ρ n + p n+. ( This is similar to equation (8, but with the cell-centered pressures at timet n+,p n+, wherep n+ is evaluated using an eplicit scheme applied to the Lagrangian form of the equation of pressure evolution in (5, which is given by p n+ = p n t (c ρ n ( u n + ρ u n ρ. ( 3.. The Lagrangian Phase Production ICE chooses the vector of conserved variables to include mass, linear momentum and internal energy. The use of the non-conservative form of the system of Euler equations in ( means that the Lagrangian part in Production ICE is given by V L U L = V n U n t p n+ p +/ p / (u +/ u /. ( Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

9 METHOD FOR COMPRESSIBLE FLOW PROBLEMS The Eulerian Phase The change in solution values due to advection over the step [t n,t n+ ] is given by V n+ U n+ = V L U L t(u + U + u U. (3 However, the numerical values of face-centered advected quantities in the following definition: U + = t tn+ U( +,tdt (4 t n has not been quantified so far in this paper and we will now show how to approimate it. Normally U( +,t is not constant for the step [t n,t n+ ], but a first-order accuracy is obtained by assuming that this is constant and is an upwinded cell-centered value. However, there are the cell-centered values at two different time levels that are available for the Eulerian phase. These are at t n and the value after the Lagrangian step. By chosing the upwinded cell-centered values at time t n for the face-centered advected quantities, we have U + = U+ n if u < + otherwise. U n Alternatively, if the upwinded cell-centered values at Lagrangian time level are considered for facecentered advected quantities, we have U + = U+ L if u < + otherwise. U L The Production ICE code uses (5 to define the face-centered advected quantities, [4, 7, 8]. (5 ( State Variables Update Phase The averages of cell primitive variables ρ,u,e, and p are then updated using the averages of cell variables ρ,ρu,ρe and the equation of state (. 4. CFL CONDITION The choice of the time step t in time integration affects the stability of the ICE method. As mentioned in [34], one requirement for the method to be stable is the fastest wave at a given time is allowed to travel, at most, one cell length in the chosen time step t. For the system of Euler equations, the time step t is chosen to satisfy the condition: t = C cfl Sma n, (7 where C cfl is a Courant or CFL coefficient satisfying < C cfl < and S n ma is the largest wave speed present through the domain at time t n. A practical choice of S n ma as mentioned in Toro [34] Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

10 L.T. TRAN, M. BERZINS is Sma n = ma( u n +c n. (8 However, Kwatra et al. [3] proposed a novel method for alleviating the stringent CFL condition imposed by the sound speed in simulating highly nonlinear compressible flow with shocks, contacts and rarefactions. It is mentioned in [3] that the maimum speed in equation (8 is too restrictive for flows where the sound speed, c, may be much larger than u, so the stringent CFL time step restriction imposed by the acoustic waves can be avoided and only the material velocity CFL restriction is used in calculating the maimum speed. The proposed method of [3] is well suited to the semi-implicit solver like the ICE method where only the advection step is the eplicit part. The proposed maimum speed calculation in [3] is: Sma n = ma u n. (9 The time step used in this paper for both Production ICE method and method is determined using (7 where S n ma is calculated using (9. 5. METHOD The method improves the numerical solutions to the one-dimensional, time-dependent Euler equations of gas dynamics by solving the conservation form in (3. However, as will be validated by numerical eperiments in Appendi C, the numerical solutions obtained using the cell-centered ICE method in conservation form have unphysical oscillations that need to be reduced or eliminated. The oscillations in the numerical solutions of the cell-centered ICE method cannot be diminished by decreasing the time step, so in this section, we will describe the algorithm used to suppress these oscillations numerically by using a simple approimate Riemann solver. 5.. Numerical Discussion To help eplain the method, we start with a discussion of schemes that approimate conservation laws as follows and consider a one-dimensional system in a conservation law form U(,t t + F(U(,t =, [a,b] and t, (3 whereu(,t is the vector of conserved variables andf(u(,t is the vector of flues. In order to approimate the solution of (3 with the initital condition U(, = U (, (3 we discretize space into N uniform cells as in Section. The cell average of the cell [, + ] at time t n is denoted by U n, where U n = + U(,t n d. (3 Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

11 METHOD FOR COMPRESSIBLE FLOW PROBLEMS A standard approach is used in integrating system (3 in space and time in the control volume [, + ] [t n,t n+ ] to give: + tn+ [ ] [U(,t n+ U(,t n ]d = F(U( +,t F(U(,t dt. t n This can then be written in the standard conservation form: U n+ ( = U n t F + F (33 where F + = t tn+ F(U( +,tdt. (34 t n Equation (33 is used by finite volume methods to solve the system (3 approimately. In order to use this relation, a spatial integration of the initital condition is required and the approimations of the flues at the cell interfaces are needed. The numerical flu derivation follows the cell-centered ICE method of Kashiwa et al. [] will be derived shortly. The system of Euler equations (3 of gas dynamics is written in the form (3 where U = (ρ,ρu,ρe T and F(U = (ρu,ρu +p,ρue +up T. The face-centered flu (34 in this case is written as F + = tn+ t tn+ t t t n (ρu( +,tdt t n (ρu +p( +,tdt tn+ t n (ρue +pu( +,tdt A Taylor series approimation of u( +,t is given by Using the notations u n+ +. (35 u( +,t = u( +,t n+ +(t t n+ u t( +,t n+ +O( t. (36 tn+ (ρu( t +,tdt = tn+ t n t = u( +,t n+ and (u t n+ = u + t ( +,t n+, we have: = u n+ + With a similar approach, we also have and [ ρ( +,t t n ( tn+ t u n+ + t n ρ( +,tdt ] +(t t n+ (u t n+ +O( t dt + +O( t. tn+ ( (ρu +p( t +,tdt = un+ tn+ (ρu( + t n t +,tdt +p n+ +O( t + t n tn+ ( (ρue +up( t +,tdt = un+ tn+ (ρe( + t n t +,tdt +(up n+ +O( t. + t n Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

12 L.T. TRAN, M. BERZINS Then face-centered flu F + vector in (35 is rewritten as F + = un+ + t n ρ( +,tdt t n (ρu( +,tdt + tn+ t n (ρe( +,tdt tn+ t tn+ t t p n+ + (up n+ + +O( t. (37 The reader should note that equation (33 with the terms F and F + are defined by equation (37 will be used in Section 7 to assess the numerical accuracy of the method. 5.. Implementation The scheme used for approimating the fluing velocity, u, in the method is similar to + equation (9, that is: u + = u n + t ( p n+ + pn+ ρ n +. (38 This equation was obtained by replacing u n ρ in (9 with u n, and ρ n with ρ n While these quantities denote the velocity and density at face-center att n, their numerical values are determined differently. While u n ρ and ρ n are determined using the weighted averages + + in ( and (, the values u n and ρ n are determined based on the simple approimate + + Riemann solver that will be discussed in Section 5.3 below. Since the pressures used in the (38 are time-advanced values, we need to perform a pressure corrector to obtain these. The eplicit pressure corrector in (6 is the one used in our implementation of the method. However, it is worth looking at the implicit pressure corrector and seeing the difference between the solutions of these two methods. By substituting ( into (7, the equation for cell-centered pressure corrector now becomes δp n = t ( p n + p n un t (c ρ n (ũ + ũ [ ] [ ] t δp n + (c ρ n + δp n ρ n δpn δpn ρ n. + Let σ = t and rearrange the terms of above equation to get [ ] +σ (c ρ n ρ n + (39 +σ (c ρ n ρ n δp n σ (c ρ n ρ n δp n + σ (c ρ n ρ n δp n (4 + ( p n + p n σ(c ρ n (ũ + ũ. = σu n Therefore, the values δp n are the solutions of the tri-diagonal linear system: A = b (4 Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

13 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 3 where: b c a b c A = a 3 b 3... c N d d b = d 3. δp δp = δp 3. (4 a N b N d N δp N and a,b,c,d are defined as follows: a = σ (c ρ n ρ n for =..(N, (43 b = +σ (c ρ n ρ n + +σ (c ρ n ρ n for =..(N, (44 d = σu n c = σ (c ρ n ρ n for =..(N, (45 + ( p n + p n σ(c ρ n (ũ + ũ for =..(N, (46 and b,b N,d and d N are obtained from the boundary condition. So in order to use the implicit pressure corrector, we have to solve the tri-diagonal linear system in (4. While this obviously takes more time to calculate than the eplicit pressure corrector in (6, the results computed using the two methods show that there is not much different between the numerical solutions obtained from using the implicit and eplicit pressure corrector in the method for the Euler equations eamples used here. The method calculates the face-centered pressure, p, the same way as the cellcentered ICE method does in Section.. It uses the calculation described in equation + (8. The method chooses to conserve the total energy instead of internal energy, so the vector of conserved variables is U = [ρ,ρu,ρe] T. The Lagrangian and Eulerian phases of the method are then given by and V L U L = V n U n t p +/ p /, (47 p +/ u +/ p / u / V n+ U n+ = V L U L t(u + U + u U, (48 in which the terms U and U + u Application of Slope Limiters in the Method L are given by equation (6 and V = V n + t(u + In common with many methods for conservative laws, slope limiters may be applied to the calculation of face-centered fluing velocity, u +/. For the face-centered fluing velocity, slope limiters are used in the estimation of face-centered quantities at t n ; in particular, they are used in Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

14 4 L.T. TRAN, M. BERZINS the calculation ofρ n andu n. This approach originates from the idea of approimating the cellcentered state by the reconstructed states obtained from the left and right cell-averaged states of + + the previous time-step. The slope limited, reconstructed states are used as inputs to a Riemann solver to determine the state at the cell interface. This will be discussed in detail below. While U is often used to denote the vector of conserved variables, the vector W, W = (ρ,u,p, is often used to denote of the vector of primitive variables, see Laney [4] and Toro [34]. LetW n be the vector of average cell-centered values of primitive variables of cell at time t n, then the value ofw on the spatial domain att n is represented by the piecewise constant data { } W n. The simplest and widely-used way to modify the piecewise constant data { } W n is to replace the constant state W n by a piecewise linear functions Wn (. The construction of the piecewise linear functions can be found in many papers; the construction in Toro [34] will be used as described below. As for the first-order Godunov method, one assumes that W n represents an integral average in cell I = [, + ] as given by A piecewise linear, local reconstruction of W n is W n = + W n (d. (49 W n ( = W n +( W n, I, (5 where W n is a suitably chosen slope of W n ( in cell I. The integral of W n ( in cell I is identical to that of W n and thus the reconstruction process is conservative. The slope Wn can be approimated by a simple finite difference formula given by W n = Wn + Wn. (5 However, to achieve a higher order scheme and to maintain bounded solutions, the slope at the current node is usually limited based on adacent slopes. The obtained slope is a limited slope W n which is used as : W n ( = W n +( W n, I, (5 to approimate W on I. The ratio r represents the ratio of successive gradients on the solution mesh at, and the limited slope W n r = Wn Wn W+ n, (53 Wn may be written in the form: W n = φ(r W n, (54 where φ(r is some flu limiter function. For the results in this paper, we choose the Monotonized Central(MC limiter function for calculating the limited slope in (54. The MC limiter function by Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

15 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 5 Van Leer [4] is φ(r = ma[,min(r,.5+.5r,]. (55 At each interface +, we now may consider the so called Generalized Riemann Problem(GRP as follows U t + F(U =, (56 W n W(,t n = ( < + W+ n ( > +, where W n ( is the limited local reconstruction in (5. Naturally, for non-linear systems the eact solution of the GRP is eceedingly complicated, but for the purpose of evaluating face-centered states, an approimate solution may be suffice. In this approach, we are not trying to evaluate the solution of the GRP in (56 analytically but rely on the boundary etrapolated values at the interfaces. The values of W n at cell boundaries using local reconstructions W + n( and Wn + ( are denoted as W n(l and W n(r where + + W n(l + = W n ( + ; Wn(R + = W+( n +. (57 The valuesw n(l andw n(r are left and right etrapolated values at the boundary at timet n. In this way, one may instead consider the conventional Riemann Problem with piecewise constant data in a new coordinate (ξ,τ where ξ = + and τ = t t n as U τ + F =, (58 ξ W n(l ξ < + W(ξ, = W n(r ξ >. The face-centered state at t n, W(,, is the value at the origin immediately after the interaction of the piecewise constant data W n(l and W n(r + where + W(, = lim τ + W(,τ. (59 By determining W(, in (ξ,τ coordinate, we have the values of face-centered states given by W n = [ρ n,u n,p n ] T at t n. There are several ways to approimate the solution to the piecewise constant data Riemann problem (58 and therefore to approimatew(,. In this paper, we use the simple approimate Riemann solver which was proposed by Harten et al. [5] and discussed in Davis [6] to approimate W(,. In order to use the approimate Riemann solver Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

16 6 L.T. TRAN, M. BERZINS described in these papers, we rewrite equation (58 as where U L and U R are obtained from W n(l + solution of (6 is given by U τ + F =, (6 ξ U L ξ < U(ξ, = U R ξ >, and W n(r + U L for /t < a L U(/t;U L,U R = U LR for a L < /t < a R for a R < /t U R respectively. The approimate Riemann where a L and a R are lower and upper bounds, respectively, for the smallest and largest signal velocity and (6 U LR = a RU R a L U L a R a L F(U R F(U L a R a L. (6 The bounds a R and a L for the Euler equations are defined in Davis [6] as: a L = u L c L, a R = u R +c R, (63 where u L, c L are the velocity and wave speed respectively obtained from U L, and u R, c R are the velocity and wave speed obtained from U R. The solution W(, in (59 is derived from the approimate solution U(;U L,U R in (6 which includes the approimations of u n and ρ n ; + + these in turn are used in equation (38 instead of using the mass-weighted quantities in equations ( and (. In summary, the face-centered fluing velocity, u, is estimated via the following steps. First, + using the local recontruction in (5, the left and right etrapolated values at this cell-center are obtained using (57. These etrapolated values then form the piecewise constant data to the Riemann problem (58. Second, this Riemann problem is solved approimately using the approach of Harten et al. [5] and Davis [6]. The approimate Riemann solution includes the approimate face-centered density,ρ n, and face-centered velocity,u n. Third, the pressure correctors,δp n + +, are calculated using equation (7. Finally, equations ( and (3 are used to calculate u NUMERICAL RESULTS AND COMPARISONS The following well-known test problems are often used to test the accuracy and robustness of many numerical methods in fluids. These tests for the one-dimensional, time-dependent Euler equations for ideal gases can be found in Toro [34] where they are used to access the performance of the numerical schemes being presented in the book and also employed here to illustrate the performance of the Production ICE method and the method. In these chosen problems, two constant states, W L = [ρ L,u L,p L ] T and W R = [ρ R,u R,p R ] T, are separated by a discontinuity Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

17 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 7 Table I. Data for one-dimensional test problems with known eact solutions, for the time-dependent, one dimensional Euler equations Problem ρ L u L p L ρ R u R p R T end P P P P P at a position =. The states W L and W R are given in Table I and the problem domain is (,t [,] (,T end ]. P is the well known Sod s problem and P is a modified version of P. These tests are considered very mild, but as mentioned in Toro [34] they are useful for assessing the entropy satisfaction property of numerical methods. P3 is considered a very hard problem for numerical methods. As mentioned in Toro [34], the solution to P4 represents the collision of two strong shocks and consists of a left facing shock, a right travelling contact discontinuity and a right travelling shock wave. Problem P5 is La s test problem [5]. Beside the problems with known eact solutions in Table I, we also include here the numerical solutions to the Shu and Osher [3] test problem. This test problem contains detailed features and structures and is considered by Greenough and Rider [8] to be a good one-dimensional surrogate for the interaction of a shock wave with a turbulent field. The initial condition at t = of the problem is defined as (3.8574,.6936, if < 4. (ρ,u,p(, = (. +.sin(5,.,. otherwise, on spatial domain [ 5.,5.]. The final time for this problem is T end =.8. As the analytical solution of this test problem is not readily available, we use the eact solution in Martin et al. [9] to show how accurate of the numerical methods in this paper. The eact solution of Shu and Osher test problem in Martin et al. [9] is obtained with the unmodified WENO-JS scheme with r = 3 and p = on 6 grid points. The numerical results for the above test problems of the method are compared against those of the Production ICE method and shown in Figures 6. A significant improvement in numerical solutions of the method is clearly shown through these figures. As shown in these figures, the profiles of the numerical solutions of Production ICE are not close to the eact solutions. This is due to the use of the non-conservative form in Production ICE. It can also be seen that, there are no eisting oscillations at shock-front in the numerical solutions of the method. The improvement in the numerical solutions of the method compared to the Production ICE method comes from the use of a conservation form in the Lagrangian phase and the application of slope limiters in the data reconstruction of the Riemann problem. In order to see how the contribution of each source into the improvement of the method, we include in Appendi C the comparision between the numerical results of the method and the cellcentered ICE method. The cell-centered ICE method denotes the method that is implemented using the cell-centered ICE method of Kashiwa et al. [] described in Section. which conserves mass, Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld (64

18 8 L.T. TRAN, M. BERZINS Production ICE Production ICE.8.8 Density.6.4 Velocity (a Production ICE 3 (b Production ICE Internal Energy.5.5 (c Pressure (d Figure. Production ICE and numerical solutions for P test problem with N=(cells and C cfl =.: (a density; (b velocity; (d internal-energy; and (c pressure. Production ICE.5 Production ICE Density (a Production ICE 3.5 Velocity.5.5 (b Production ICE Internal Energy (c Pressure (d Figure. Production ICE and numerical solutions for P test problem with N=(cells and C cfl =.: (a density; (b velocity; (d internal-energy; and (c pressure. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

19 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 9 Production ICE 5 Production ICE 8 Density 6 4 Velocity 5 5 Internal Energy (a Production ICE (c Pressure 5 (b Production ICE (d Figure 3. Production ICE and numerical solutions for P3 test problem with N=8(cells and C cfl =.: (a density; (b velocity; (d internal-energy; and (c pressure. 5 Production ICE Production ICE 4 5 Density 3 Velocity 5 5 Internal Energy (a Production ICE (c Pressure (b Production ICE 5 5 (d Figure 4. Production ICE and numerical solutions for P4 test problem with N=(cells and C cfl =.: (a density; (b velocity; (d internal-energy; and (c pressure. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

20 L.T. TRAN, M. BERZINS.5 Production ICE.5 Production ICE Density.5 Velocity.5 (a Production ICE (b Production ICE Internal Energy 5 5 Pressure 3 (c (d Figure 5. Production ICE and numerical solutions for P5 test problem with N=(cells and C cfl =.: (a density; (b velocity; (d internal-energy; and (c pressure. Density Production ICE " Solution" (a Velocity Production ICE " Solution" (b Figure 6. Production ICE and numerical solutions for Shu and Osher test problem with N=6(cells and C cfl =.: (a density and (b velocity. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

21 METHOD FOR COMPRESSIBLE FLOW PROBLEMS linear momentum and total energy. From the results in Figures 6 and Figures 5, it may be seen that the use of conservation form improves the solution profiles and the reconstruction of the Riemann problem with the slope limiters helps to eliminate the non-physical oscillations. 7. ACCURACY IN SPACE AND TIME There are two sources of errors in the numerical solutions of the method: spatial and temporal errors. The spatial error comes from the spatial discretization of the PDEs and the temporal error comes from the time integration method. Let e U (t n be the total error in approimating U n of cell at t n which is given by e U (t n = + U(,t n d U n. (65 [ ] Let U tn ;t,u be the eact solution of discretized system with the initial condition U (t = U. We then have e U (t n = es U (t n +et U (t n, (66 where es U (t n = + [ ] U(,t n d U tn ;t,u (67 and [ ] et U (t n = U tn ;t,u U n (68 are the spatial error and the temporal error respectively. 7.. Temporal Error Let le U (t n be the time local error of the step [t n,t n ] of the method which is defined by [ ] le U (t n = U tn ;t n,u n U n, (69 [ ] and where from equation (33, the eact local solution U tn ;t n,u n is given by [ ] U tn ;t n,u n = U n t ( F + F, (7 where F + is defined in (37. On the other hand, the solution at t n is U n = U n t ( F + F, (7 where F + = u + U + + p + u p + +. (7 Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

22 L.T. TRAN, M. BERZINS Therefore le U (t n = t [( F + From equations (37 and (7, we have: F + F + = u n+ + As + ( t p n+ + (up n+ + tn F + ( F U( +,tdt U + t n p + u p + + F ]. (73 ( + u n+ u + + U + +O( t. (74 ( ( u n+ u = O( t and p n+ + + p = O( t, we then have + + F + F + = u n+ + ( t tn By considering the epansion of U( +,t about U + : ( U( +,t = U + +(t t n U + equation (75 now becomes Therefore ( F + F + F + F + = t ( F F t n U( +,tdt U + un+ + ( = t t + (t t n U + t ( [ ( u n+ U + + +O( t. U + +O( t. (75 tt +..., (76 +O( t. (77 t ( u n+ U From equations (73 and (78, le U (t n is second-order in t for a fied. Therefore, et U (t n is first-order in t. As U is a vector of conserved variables, et U (t n is also a vector of temporal [ T errors of these conserved variables where et U (t n = et ρ (t n,et ρu (t n,et ρe (t n ]. Define the approimate L-error norm of the vector of temporal errors for variable q, where q = ρ,ρu,ρe,u, or p, as follows et q (t n L = t ] (78 N et q (t n, (79 = where et q (t n = [et q (t n,et q (t n,et q 3 (t n,...,et q N (t n] T. In order to calculate the temporal error in equation (68, we need to determine the timeintegrated eact solution U Tend ;t,u in this equation. As we do not have the eact solution [ ] [ ] U Tend ;t,u, we assume that the calculated solutionu n converges to the time-integrated eact [ ] solution U Tend ;t,u when reducing Ccfl. Therefore, we use a highly resolved solution as Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld

23 METHOD FOR COMPRESSIBLE FLOW PROBLEMS 3 Table II. Temporal Error:L -norms and the order of accuracynof the conserved and primitive [ variables for the test cases in Table I using N=(cells. The time-integrated eact solutions U Tend ;t,u ] for the discretized problems of these test cases are obtained by using C cfl =.. The notation ae-b used here stands for a b. et ρ (T end et ρu (T end et ρe (T end et u (T end et p (T end C cfl. L n. L n. L n. L n. L n.4 6.9E-4 5.7E-4.6E-3.5E-3 4.4E-4 P. 3.37E E E E-4.3.E E-4..36E-4..76E-4..55E-4..5E E E-5..39E-4..7E E E-3.4E-3 3.5E-3.98E-3.E-3 P. 7.55E E E E E E E E E-4..76E E-4..7E E-4..46E-4..38E E-.8E- 3.44E- 3.E- 6.6E- P3. 5.E-3.6.4E-.6.63E-.8.53E-.3 3.E E E-. 8.6E E-3..53E-..5.7E-3..55E-. 4.E-. 3.8E E E- 3.4E- 5.6E-.85E-.5E- P4..94E-..59E-.3.59E E E E E-..6E E E E E-. 6.E-.5.3E E E E-3 6.3E-3.39E-3.64E-3 P5..4E-3..75E-3. 3.E E E E E-4..6E E E E E E-4..83E-4..5E-4. [ ] the time-integrated eact solution U Tend ;t,u with Ccfl =.. This solution meets the criterion mentioned in Greenough and Rider [8] that the grid converged solutions should be at least 8 times finer than the finest grid eamined for error. The temporal error norms and their orders of accuracy of the conserved and primitive variables for the above test cases att end are shown in Table II. The results in Table II show that the orders of accuracy of the conserved variables for these test cases are very close to, this is consistent with the above analysis. The orders of accuracy of the primitive variables are also very close to. 7.. Spatial Error With the linear spatial discretization as discussed above, the spatial error of the vector of conserved variables U is first-order in. In order to access the spatial errors of a test case, we need the [ ] eact solution U Tend ;t,u, see equation (67. The result in Section 7. gives the rate at which the computed solution approaches the true solution, so a more accurate approimation to the eact [ ] solution U Tend ;t,u might be obtained by comparing the numerical solution to a finer-mesh [ ] numerical solution. Therefore, we estimate the eact solution U Tend ;t,u in Table III using the computed solutions of the method with C cfl =.5 and one with C cfl =.5. Define the L-error norm of the vector of spatial errors for variable q, where q = ρ,ρu,ρe,u, or p, as follows es q (t n L = N es q (t n (8 Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld =

24 4 L.T. TRAN, M. BERZINS Table III. Spatial Error: L -norms and the order of accuracy [ m of the conserved and primitive variables for the test cases in Table I. The eact solutions U Tend ;t,u ] are the converged numerical solutions discussed in Section 7.. es ρ (T end es ρu (T end es ρe (T end es u (T end es p (T end N. L m. L m. L m. L m. L m.38e-.5e-.68e-.5e-.6e- 9.6E E E-.79.E E-3.79 P E E E E E E E E E E E E E E-3.9.9E E-.89E- 4.47E-.83E-.66E-.45E-.58.6E E E-.8.E-.7 P E E-3.5.8E E E E E-3.58.E E E E E E E-3.7.7E-3.7.8E- 3.49E+ 8.6E+ 5.56E-.6E+.47E-.9.96E E E E+.74 P3 4.9E-.43.E E E E E-.5.5E+.55.6E+.6.6E E E-.48.7E+.49.9E E-.9.5E E- 6.8E+ 6.93E+.E-.73E+ 5.76E-.4 5.E E E E+.97 P E E E E E E E E E-..9E+. 6.9E-.5.67E E+.66.4E-.8.9E E- 6.47E-.53E- 3.4E- 3.35E- 3.E E E-.87.8E-.74.E-.68 P5 4.3E E E-.69.E-.86.6E E-.53.6E E E E E E-.55.7E E E-3.86 where es q (t n = [es q (t n,es q (t n,es q 3 (t n,...,es q N (t n] T. The spatial error norms and their orders of accuracy for the above test cases at T end are shown in Table III. Theoretically the spatial error order of accuracy is first-order. However it is shown in Table III that the order of accuracy is mostly below due to the discontinuities in the solutions of these test cases. Greenough and Rider [8] mention earlier work showing less than first-order accuracy for the first-order version of Godunov s method and suggest that this is due to the low resolution computed solutions being very different from the highly resolved solution. For reference purposes, we include in Appendices A and B of this paper the spatial errors and the orders of accuracy for the inviscid and viscous Burgers problems. For the inviscid Burgers problem, the order of accuracy is below. For the viscous Burgers problem, the order is around. As numerical solutions obtained with first-order methods are diffusive and not accurate enough to be used for some large problems on relatively coarse grids; for eample, the numerical solution to Shu and Osher test problem shown in Figure 6, we improve the order of accuracy of the method to nd-order in both space and time in the following sections. Copyright c John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids ( DOI:./fld