Numerical study on the convergence to steady state solutions of a new class of finite volume WENO schemes: triangular meshes Jun Zhu and Chi-Wang Shu Abstract In this paper we continue our research on the numerical study of convergence to steady state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in [8], from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in [7] for the third and fourth order version and in this paper for the fifth order version, with the main idea being the application of constructing one high degree polynomial on a big central stencil to get high order approximation in smooth regions and several linear polynomials defined on small stencils located centrally and in different sectorial regions which are partitioned by the barycenter and three vertices of the triangular cells to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in [8], by performing such spatial reconstruction procedures together with a TVD Runge-Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations which are responsible for the residue of classical WENO schemes to hang at a truncation error level instead of converging to machine zero. The third, fourth and fifth order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady state simulations in our extensive numerical experiments. Key Words: WENO scheme, triangular mesh, finite volume scheme, steady state solution, convergence property. AMS(MOS) subject classification: 6N8, L6 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 6, P.R. China. E-mail: zhujun@nuaa.edu.cn. Research supported by NSFC grant 7 and the State Scholarship Fund of China for studying abroad. Division of Applied Mathematics, Brown University, Providence, RI 9, USA. E-mail: shu@dam.brown.edu. Research supported by ARO grant W9NF- and NSF grant DMS-794.
Introduction In this paper, we continue our numerical study on the convergence to steady state solutions of the Euler equations { f(u)x + g(u) y =, u(x, y) = u (x, y) (.) in which u = (ρ, ρµ, ρν, E) T, f(u) = (ρµ, ρµ +p, ρµν, µ(e +p)) T and g(u) = (ρν, ρµν, ρν + p, ν(e +p)) T, with ρ being the density, µ and ν being the velocities in the x and y directions, respectively, p being the pressure and E = p γ + ρ(µ + ν ) being the total energy where γ =.4 for air. We initialized this line of study in [8], in which the one-dimensional case and two-dimensional case with tensor product meshes are considered. In this paper we would like to consider the case of unstructured triangular meshes. As before, we would like to consider the approach of first solving the unsteady Euler equations { ut + f(u) x + g(u) y =, u(x, y, ) = u (x, y), (.) with some suitable time discretization methods. The numerical solution of the steady state equations (.) is then obtained when the time derivative or the residue of the unsteady Euler equations (.) approaches machine zero. We refer to our earlier paper [8] and the references therein for a brief survey of high order numerical methods. We consider weighted essentially non-oscillatory (WENO) finite volume schemes in this paper. WENO schemes for structured meshes were designed in [, ] and for unstructured meshes in [, 4, 9,, ]. We would like to mention in particular the two dimensional finite volume WENO schemes on triangular meshes proposed by Hu and Shu in [4]. In their paper, they proposed a third order WENO scheme using a combination of nine linear polynomials and a fourth order WENO scheme using a combination of six quadratic polynomials, and gave a new way of measuring two dimensional smoothness of numerical solutions which was different from the expressions specified in [, ]. However, the procedure of obtaining the optimal linear weights for the third order and fourth order finite volume WENO schemes in [4] was quite complicated,
and the positivity of the linear weights cannot be guaranteed, thus requiring additional techniques to treat negative linear weights [4] to ensure the non-oscillatory property. A major difficulty for using a time-dependent high order solver to march to steady state is to drive down the residue to machine zero. When applying the classical finite difference and finite volume WENO schemes [4,, 6, ] with the third order TVD Runge-Kutta time discretization [,, 7] for solving the steady state problems, the residue often hangs at the truncation error level which is far above the machine zero. We again refer to our earlier paper [8] and the references therein for a survey of earlier attempts in the literature to deal with this difficulty. In [8], we studied numerically the convergence behavior to steady states of a new class of finite difference and finite volume WENO schemes on structured meshes for hyperbolic conservation laws. This new class of finite difference and finite volume WENO schemes was designed for unsteady hyperbolic conservation laws in [, 6, 7], see also related work in [, 4, 6,, 6, 8, 9, ]. This new class of WENO schemes uses a convex combination of a high order polynomial with two linear polynomials on unequal sized spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension manner. Such WENO schemes use the same information as the classical WENO schemes and yield the same formal order of accuracy in smooth regions, yet they have a few advantages over the classical WENO schemes, one of them being that the residue of these new WENO schemes can settle down to a small number close to machine zero for an extensive list of the standard test problems, as demonstrated in [8]. In this paper, we extend the result in [8] to two-dimensional triangular meshes. We consider both the third order and fourth order finite volume WENO schemes in [7] and a new fifth order WENO scheme, defined on triangular meshes, and apply all three finite volume WENO schemes to the solution of steady state problems. These WENO schemes use a convex combination of one quadratic, cubic or quartic polynomial and four linear polynomials on unequal sized spatial stencils on triangular meshes. The third order and
fourth order WENO schemes use the same information as the classical WENO schemes in [4] and yield the same formal order of accuracy in smooth regions, yet they have a few advantages over the classical ones. One of them is that the new WENO schemes can provide a unified polynomial approximation in the whole triangular cell with the same linear and nonlinear weights, comparing with the classical WENO schemes [4] which involve different linear and nonlinear weights for different quadrature points inside the triangular cell. Also, it appears that the fifth order finite volume scheme constructed in this paper is the first time that a fifth order finite volume WENO scheme with only five spatial stencils containing sixteen distinct cells on triangular meshes has ever been constructed. We study the performance of these finite volume WENO schemes for steady state problems on triangular meshes, and demonstrate that the residue could settle down to a small number close to machine zero for all our test cases. To the best of our knowledge, this appears to be the first time that such high order finite volume WENO schemes could have residue settling down to machine zero with standard Runge-Kutta time discretization [,, 7] on triangular meshes. Of course, other time marching methods as well as special tools such as preconditioning to speed up steady state convergence could make the steady state convergence more efficient, however this is not our main focus and hence will not be explored in this paper. The organization of the paper is as follows. In Section, we give the framework of the third order and fourth order finite volume WENO schemes as developed in [7], and propose a new fifth order WENO scheme for solving hyperbolic conservation laws on triangular meshes. In Section, an extended list of classical steady state problems are presented to verify the good performance of the new WENO schemes in the residue convergence to machine zero. Concluding remarks are given in Section 4. 4
Finite volume WENO schemes on triangular meshes We consider the two dimensional conservation laws (.) on triangular meshes and integrate (.) over the target cell to obtain the semi-discrete finite volume formula dū (t) dt = F nds = L(u), (.) in which ū (t) = u(x, y, t)dxdy, F = (f, g), is the boundary of the target cell, is the area of and n is the outward unit normal to the boundary of. The line integrals in (.) are discretized by a two-point (for the third order and fourth order schemes) or three-point (for the fifth order scheme) Gaussian integration formula [4] on every edge F nds ll ll= r σ l F(u(x Glll, y Glll, t)) n ll, (.) l= and F(u(x Glll, y Glll, t)) n ll, l =,..., r (r= for the third and fourth order schemes or r= for the fifth order scheme), ll =,, are replaced by numerical fluxes such as the Lax-Friedrichs flux F(u(x Glll, y Glll, t)) n ll [(F(u+ (x Glll, y Glll, t)) + F(u (x Glll, y Glll, t))) n ll α(u + (x Glll, y Glll, t) u (x Glll, y Glll, t))], l =,..., r, ll =,,. (.) Here α is taken as an upper bound for the eigenvalues of the Jacobian in the n ll direction, u and u + are the reconstructed values of u defined inside and outside of the target cell at different Gaussian quadrature points, and ll, ll =,, are the lengths of the line segments. Next, we describe the spatial reconstructions of the third order and fourth order finite volume WENO schemes in [7] and their extension to a new fifth order WENO scheme on triangular meshes. For simplicity, five unequal sized spatial stencils containing enough distinct triangular cells for different orders of WENO schemes are relabeled in Figure.. The reconstruction of the function u(x, y, t) at different Gaussian quadrature points (x Glll, y Glll ), l =,..., r, ll =,, on the boundaries of the target cell are speci-
L L L Figure.: The spatial stencils for reconstructions of different orders with three sectorial regions. From top to bottom and left to right: T = {,,,,,,,,, } (for the third and fourth order reconstructions), or T = {,,,,,,,,,,,,,,, } (for the fifth order reconstruction); T = {,,, }, T = {,,, }, T 4 = {,,, }, T = {,,, }. 6
fied as follows. We suppress the variable t in u(x, y, t) in the following when there is no confusion. Step. Select a big central spatial stencil T containing enough distinct triangular cells and obtain high degree polynomials for reconstructions of different orders (see Figure.). Step.. For the third order reconstruction, select a big stencil T = {,,,,,,,,, }. Then we construct a quadratic polynomial p (x, y) span{, x x, y y, (x x ), (x x )(y y ), (y y ) } on T to obtain a third order approximation of the conservative variable u, in which (x, y ) is the barycenter of the target cell. Such quadratic polynomial has the same cell average of u on the target cell and matches the cell averages of u on the other triangles in the set T \ { } in a least square sense [4]: p (x, y)dxdy = ū, p (x, y) = argmin ( p(x, y)dxdy ū l ),(.4) l l A l A = {,,,,,,,, }. Step.. For the fourth order reconstruction, select a big stencil T = {,,,,,,,,, }. Then we construct a cubic polynomial p (x, y) span{, x x, y y, (x x ), (x x )(y y ), (y y ), (x x ), (x x ) (y y ), (x x )(y y ), (y y ) } on T to obtain a fourth order approximation of the conservative variable u by requiring that it has the same cell averages of u on all triangles: p (x, y)dxdy = ū l, l =,,,,,,,,,. (.) l l Step.. For the fifth order reconstruction, select a big stencil T = {,,,,,,,,,,,,,,, }. Then we construct a quartic polynomial p (x, y) span{, x x, (x x ) (y y ), (x x )(y y ) y y, (x x ), (x x )(y y ), (y y ), (x x ), (y y ), (x x ) 4, (x x ) (y y ), (x x ) (y y ), (x x )(y y ), (y y ) 4 } on T to obtain a fifth order approximation of the conservative variable u by requiring that it has the same cell average of u on the target cell and matches the cell averages of u on, 7
the other triangles in the set T \ { } in a least square sense [4]: p (x, y)dxdy = ū, p (x, y) = argmin ( p(x, y)dxdy ū l ),(.6) l l A l A = {,,,,,,,,,,,,,, }. Step. Connect the barycenter of the target cell with its three vertices and define three lines L, L and L [] (see Figure.). Then the field is split into three sectors. Every sectorial stencil consists of the target cell and its neighboring cells and the barycenters of such cells should lie in the same sector. Such three sectorial stencils T = {,,, }, T = {,,, }, T 4 = {,,, } and one small central stencil T = {,,, } are chosen, respectively. Then four linear polynomials on such sectorial and central stencils [, 8, ] are defined. We can obtain such linear polynomials y y p l (x, y) span{, x x, } on T l, l =,...,, to obtain second order approximations of the conservative variable u by requiring that they have the same cell averages of u on the target cell and match the cell averages of u on the other triangles in a least square sense [4]: p κ (x, y)dxdy = ū, p κ (x, y) = argmin ( p(x, y)dxdy ū l ),(.7) l l A κ l κ =,..., ; A = {,, }, A = {,, }, A 4 = {,, }, A = {,, }. Step. Define the linear weights. With the similar idea for CWENO schemes proposed by Levy, Puppo and Russo [8, 9], we have ( p (x, y) = γ p (x, y) γ l= γ l γ p l (x, y) ) + γ l p l (x, y). (.8) Note that (.8) holds true for any choice of the linear weights γ l, l =,..., on the condition that γ. Following the practice in [9, 4,, 9], we take the positive linear weights as γ =.96 and γ =γ = γ 4 =γ =. in this paper. Step 4. Compute the smoothness indicators, denoted by β l, l =,...,, which measure how smooth the functions p l (x, y), l =,..., are in the target cell. The smaller these 8 l=
smoothness indicators, the smoother the functions are in the target cell. We use the same recipe for the smoothness indicators as in [4, ]: β l = r ( ) l l x l y l p l (x, y) dxdy, l =,...,, (.9) l = where l = (l, l ), l = l + l. r =,, 4 for l =, and r = for l =,...,, respectively. Step. Compute the nonlinear weights based on the linear weights and the smoothness indicators. We use τ [, 6, 7] for unequal degree polynomials defined on unequal sized spatial stencils which is simply defined as the absolute deference of pairs in β l, l =,..., between large and small stencils. We adopt the strategy in WENO-Z as specified in [,, 8] and define ( ) β β + β β + β β 4 + β β τ =. (.) 4 The nonlinear weights are then defined as ω l = ω l κ= ω, ω l = γ l ( + τ ), l =,...,. (.) κ ε + β l Here ε is a small positive number to avoid the denominator of (.) to become zero. We take ε = 6 in all steady state simulations in this paper. Step 6. The final reconstruction polynomial for the approximation of u(x, y, t) at any points inside the target cell is given as ( u(x, y, t) ω p (x, y) γ κ= γ κ γ p κ (x, y) ) + ω κ p κ (x, y). (.) κ= Step 7. The third order TVD Runge-Kutta time discretization [,, 7] u () = u n + tl(u n ), u () = 4 un + 4 u() + 4 tl(u() ), u n+ = un + u() + tl(u() ), (.) is used to solve (.). Finally, we obtain fully discrete third, fourth and fifth (spatial) order WENO schemes on triangular meshes, respectively. If the number of triangular cells located inside the big central stencil T is less than six (for the third order scheme), ten (for the fourth order scheme) or fifteen (for the fifth order 9
scheme) because some of the triangles coincide with each other, neighboring triangular cells in the next layer are used to reconstruct the suitable degree polynomials in a least square sense [4]. On the one hand, we should emphasize that there should be only one small spatial stencil located inside each sectorial region, as well as a central small stencil, to build a linear reconstruction polynomial for each small stencil. This appears to be responsible both for the non-oscillatory performance (without degrading high order accuracy in smooth regions) and for the residue to settle down to machine zero for steady state computation. Numerical results In this section we provide some benchmark steady state examples for demonstrating the good performance of the new third, fourth and fifth order WENO schemes on triangular meshes described in section. We take the CFL number as.6. The linear weights are set as γ =.96 and γ =γ = γ 4 =γ =. for different WENO schemes. For the system of the compressible Euler equations, the reconstructions are performed in the local characteristic directions to avoid spurious oscillations, and we refer the readers to [6, 4] for more details. The average residue is defined as Res A = N i= R i + R i + R i + R4 i, (.) 4 N where R i are the local residuals of different conservative variables, namely R i = ρ t i = ρ n+ i ρ n i t, R i = (ρµ) t i = (ρµ)n+ i (ρµ) n i t and N is the total number of cells., R i = (ρν) t i = (ρν)n+ i (ρν) n i t, R4 i = E t i = En+ i Ei n, t Example.. We solve the two dimensional Euler equations with source terms ρ ρµ ρν.4 cos(x + y) ρµ t ρν + ρµ + p x ρµν + ρνµ y ρν + p =.6 cos(x + y).6 cos(x + y). (.) E µ(e + p) ν(e + p).8 cos(x + y) The initial conditions are ρ(x, y, ) = +. sin(x + y), µ(x, y, ) =, ν(x, y, ) =, p(x, y, ) = +. sin(x + y). The computational domain is (x, y) [, π] [, π], and
6. 4. 4........ 4 4.. 6 Figure.: D Euler equations with source terms. Sample mesh. Log(Res A) - 4 Log(Res A) - 4 Log(Res A) - 4 Figure.: D Euler equations with source terms. The evolution of the average residue. From left to right: the third order, fourth order and fifth order WENO schemes. Different numbers indicate different mesh levels of boundary points uniformly distributed from π to π 8. the exact steady state solution is applied as boundary conditions in both directions. For this test case, a sample mesh is shown in Figure.. The exact solution of density is ρ(x, y, t) = +. sin(x + y). The L and L errors and orders of accuracy for density at steady state are listed in Table., from which we can see that the optimal order of accuracy is achieved for different finite volume WENO schemes. The history of the residue (.) as a function of time is shown in Figure., in which we can see that the residue settles down to tiny numbers from. to for the different WENO schemes. Example.. Shock reflection problem. The computational domain is a rectangle of length
Table.: D Euler equations with source terms. Initial data ρ(x, y, ) = +. sin(x + y), µ(x, y, ) =, ν(x, y, ) =, and p(x, y, ) = +. sin(x + y). Steady state. L and L errors for density. WENO schemes. WENO WENO4 h L error order L error order L error order L error order π 4.E 9.E.9E.E π.7e.8.8e.8 7.74E.9.9E- 4. π.e-.7.7e-.64.e- 4.8.E.98 π.76e.89 8.4E.77.67E 4.7 7.8E.94 4 π.8e-.9.e.9 8.7E 4. 4.7E-7 4. 8 WENO h L error order L error order π.4e.8e π 4.9E 4.6.6E- 4. π.7e- 4.97.9E- 4.97 π.e-7 4.9.4E.7 4 π.6e 4.97.94E 4.7 8 4 and height. The boundary conditions are that of a reflection condition along the bottom boundary, supersonic outflow along the right boundary and Dirichlet conditions on the other two sides: (ρ, µ, ν, p) T ) = { (.,.9,,./.4) T (,y,t) T, (.69997,.694,.6,.89) T (x,,t) T. The sample mesh is shown in Figure.. Initially, we set the solution in the entire domain to be that at the left boundary. We show the equally spaced density contours of the third order, fourth order and fifth order WENO schemes in Figure.4 after steady state is reached. The history of the residue (.) as a function of time is also shown in Figure.4. It can be seen that the average residue of the third order, fourth order and fifth order WENO schemes can settle down to a value around., close to machine zero. In order to explore the reason of such new WENO schemes in residue time history, we plot the nonlinear weights in each of the four local characteristic fields at Gaussian quadrature points on the boundaries of the triangular cells along the line y =.. When numerical steady state has been reached, we plot the nonlinear weights in Figure. to Figure.7, in
. 4 Figure.: The shock reflection problem. Sample mesh. which we can clearly observe that the nonlinear weights can settle down to the linear weights except for a narrow region near the shocks. Example.. A Mach wind tunnel with a forward-facing step. This model problem is originally from []. The setup of the problem is as follows. The wind tunnel is length unit wide and length units long. The step is. length units high and is located.6 length units from the left-hand end of the tunnel. The problem is initialized by a right-going Mach flow. Reflective boundary conditions are applied along the wall of the tunnel and inflow/outflow boundary conditions are applied at the entrance/exit. A sample mesh is shown in Figure.8. The results are shown in Figure.9 when the numerical solution has settled down to steady state. It can also be seen in Figure.9 that the average residue of such WENO schemes can settle down to a value around., close to machine zero. Example.4. The subsonic flow past a circular cylinder at a Mach number of M =.8 []. The sample mesh is shown in Figure.. The radius of the cylinder is., and the computational domain is {(x, y) :. x + y }. The number of points on the inner and outer boundaries is 8. Mach number contours are shown in Figure.. The average residue of such WENO schemes is shown in Figure. and it can settle down to a value around., close to machine zero. Example.. We consider an inviscid Euler subsonic flow past a single NACA airfoil configuration [7] with Mach number M =.8, angle of attack α =., and with M =.8, angle of attack α =. The computational domain is [, ] [, ].
. 4. 4. 4 Log(Res A) Log(Res A) Log(Res A) - - - 4 6 8 4 6 8 4 6 8 Figure.4: The shock reflection problem. From top to bottom and left to right: equally spaced density contours from.4 to.6 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes. Boundary points are uniformed distributed with h =. 4
.8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4 4 4 4 4.8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4 4 4 4 4.8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4 4 4 4 4 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 Figure.: The shock reflection problem. The third order WENO scheme. Different signs and lines correspond to nonlinear weights of different stencils. From top to bottom: the nonlinear weights of the first, second, third and fourth local characteristic components of conservative variables of the cells along the line y =.. From left to right: the Gaussian quadrature point on the first boundary, second boundary and third boundary of the cells. Boundary points are uniformed distributed with h =. 4 444444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444 4444 444444444444444444444444444 44444444444444444444 44444 4444444444444444444444444444444444444444444444444444444444444444 4444 4 44444444444444444444444444 4444444444444444444444 4 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4444 444444444444444444444444444 4 444444444444444444444444 44444444444444444444444444444444444444444444444444444444444444444 4 44444444444444444444444444444 4444444444444444444444 4 44 4 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 44444444444444444444 44444 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444 4 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 444444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444 4 444444444444444444444444444444 4444444444444444444444 4 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4444 444444444444444444444444444 Figure.6: The shock reflection problem. The fourth order WENO scheme. Different signs and lines correspond to nonlinear weights of different stencils. From top to bottom: the nonlinear weights of the first, second, third and fourth local characteristic components of conservative variables of the cells along the line y =.. From left to right: the Gaussian quadrature point on the first boundary, second boundary and third boundary of the cells. Boundary points are uniformed distributed with h =.
.8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4.8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4.8.6.4..8.6.4..8.6.4..8.6.4. 4 4 4 4 4 4 444444444444444444444444 44444444444444444444444444444444444444444444444444444444444444444 44 44444444444444444444444444 4 444444444444444444444444 4 44444444444444444444444444444444444444444444444444444444444444444 4 4 44444444444444444444444444 4 4444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4444444444444444444444444444 4 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 4 44444444444444444444444444 444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444 4 4444444444444444444444444444 4 4 4444444444444444444444 4 444444444444444444444444444444444444444444444444444444444444444 4 4 44444444444444444444444444444 4 444444444444444444444444 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444 4 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 4 444444444444444444444444 444444444444444444444444444444444444444444444444444444444444444444 4 444444444444444444444444444 4 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4444444444444444444444444444 4 444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 44 44444444444444444444444444 Figure.7: The shock reflection problem. The fifth order WENO scheme. Different signs and lines correspond to nonlinear weights of different stencils. From top to bottom: the nonlinear weights of the first, second, third and fourth local characteristic components of conservative variables of the cells along the line y =.. From left to right: the Gaussian quadrature point on the first boundary, second boundary and third boundary of the cells. Boundary points are uniformed distributed with h =.. Figure.8: Forward step problem. Sample mesh. 6
... Log(Res A) Log(Res A) Log(Res A) - - - 4 6 8 Figure.9: Forward step problem. From top to bottom and left to right: equally spaced density contours from. to 6. of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes. Boundary points are uniformed distributed with h =. 7
Figure.: Subsonic cylinder problem. Sample mesh. Left: the whole region; Right: zoomed near the cylinder. - - - - - - Log(Res A) - Log(Res A) - Log(Res A) - Figure.: Subsonic cylinder problem. From left to right and top to bottom: equally spaced Mach number contours from.4 to.94 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes. 8 points are uniformly distributed on the inner and outer boundaries. 8
Figure.: NACA airfoil sample mesh. Left: the whole region; Right: zoomed near the airfoil. The sample mesh is shown in Figure., which consists 94 triangles. equally spaced pressure contours are shown in Figure. and Figure.4, respectively. It can be observed that the average residue of such WENO schemes settles down to a value around, close to machine zero. Example.6. Following above example, we now consider an inviscid Euler supersonic flow past a single NACA airfoil configuration [7] with Mach number M =, angle of attack α =, and with M =, angle of attack α =. The computational mesh used is also shown in Figure. with the same number of triangles. The different order WENO schemes are used in the simulations. and 6 equally spaced pressure contours are shown in Figure. and Figure.6, respectively. It can be observed in these two figures that the average residue of the new WENO schemes settles down to a value around 4. to 4, close to machine zero. Example.7. We consider inviscid Euler transonic flow past a single NACA airfoil configuration [7] with Mach number M =.8, angle of attack α =. ; with M =.8, angle of attack α = ; and with M =, angle of attack α =. The computational domain is [ 6, 6] [ 6, 6]. The sample mesh containing 9 triangles is shown in Figure.7. 9
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - 4 6 8 4 6 8 4 6 8 Figure.: NACA airfoil. M =.8, angle of attack α =.. From left to right and top to bottom: equally spaced pressure contours from. to.46 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes.
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - 4 6 8 4 6 8 Figure.4: NACA airfoil. M =.8, angle of attack α =. From left to right and top to bottom: equally spaced pressure contours from.49 to.4 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes.
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - Figure.: NACA airfoil. M =, angle of attack α =. From left to right and top to bottom: equally spaced pressure contours from.76 to. of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes.
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - Figure.6: NACA airfoil. M =, angle of attack α =. From left to right and top to bottom: 6 equally spaced pressure contours from.76 to. of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes.
Figure.7: NACA airfoil sample mesh. Left: the whole region; Right: zoomed near the airfoil. Equally spaced pressure contours are shown in Figure.8 to Figure.. It can also be seen in these three figures that the average residue of three different order WENO schemes settles down to a value around 4 to 6, respectively, close to machine zero once again. 4 Concluding remarks In this paper we consider the third and fourth order finite volume WENO schemes on triangular meshes introduced in [7], and propose a new fifth order WENO scheme along the same lines, to numerically study their performance for solving steady state problems. In the spatial reconstruction procedure, we apply only five unequal sized stencils and reconstruct one quadratic, cubic or quartic degree polynomial and four linear polynomials. The crucial objective is to use a high degree polynomial to obtain high order approximation in smooth regions and use the WENO procedure to rely more on at least one of four linear polynomials to keep the essentially the non-oscillatory property near discontinuities. The total number of polynomials used is thus much fewer than other finite volume WENO schemes in the literature for the same order of accuracy, and the choice of the linear weights has a lot of flexibility, thus leading to robustness and efficiency of the new WENO schemes. One 4
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - 4 6 8 4 6 8 4 6 8 Figure.8: NACA airfoil. M =.8, angle of attack α =.. From left to right and top to bottom: equally spaced pressure contours from.67 to.4 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes.
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - 4 6 8 4 6 8 4 6 8 Figure.9: NACA airfoil. M =.8, angle of attack α =. From left to right and top to bottom: equally spaced pressure contours from. to. of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes. 6
- - - - - - Log(Res A) - Log(Res A) - Log(Res A) - Figure.: NACA airfoil. M =, angle of attack α =. From left to right and top to bottom: 6 equally spaced pressure contours from.6 to.7 of the third order, fourth order and fifth order WENO schemes; and the evolution of the average residue of the third order, fourth order and fifth order WENO schemes. 7
advantage of these new WENO schemes explored in this paper, following our earlier work in [8] for structured meshes, is that the residue when computing steady state problems using standard Runge-Kutta time discretization can settle down to a tiny number close to machine zero, also for the unstructured triangular meshes considered in this paper. Standard benchmark steady state examples are given to indicate the good performance of these new WENO schemes. When plotting the nonlinear weights for the shock reflection problem at steady state, we find that they are very close to the linear weights except for a very small region near discontinuities, which may explain the nice convergence of residue to machine zero. The results indicate that the new high order finite volume WENO schemes in [7] and this paper have a good potential in steady state simulations on triangular meshes in applications. In future work, we would like to explore the generalization of these results to three dimensional unstructured meshes. References [] R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation, J. Comput. Phys., 4 (99), 4-8. [] D.S. Balsara, S. Garain and C.-W. Shu, An efficient class of WENO schemes with adaptive order, J. Comput. Phys., 6 (6), 784. [] R. Borges, M. Carmona, B. Costa and W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 7 (8), 9-. [4] G. Capdeville, A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes, J. Comput. Phys., 7 (8), 977. [] M. Castro, B. Costa and W.S. Don, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys., (), 766-79. 8
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