Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter

Transcription

1 Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter Jun Zhu, inghui Zhong, Chi-Wang Shu 3 and Jianxian Qiu 4 Abstract In this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge- Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [3]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion while the simple WENO limiter [3] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. This modification improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter. Key Words: Runge-Kutta discontinuous Galerkin method; HWENO limiter; conservation law AMS(MOS) subject classification: 65M6, 35L65 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 6, P.R. China. zhujun@nuaa.edu.cn. Research supported by NSFC grants 375, 93 and 7. Department of Mathematics, Michigan State University, East Lansing, MI 4884, USA. zhongxh@math.msu.edu. 3 Division of Applied Mathematics, Brown University, Providence, RI 9, USA. shu@dam.brown.edu. Research supported by DOE grant DE-FG-8ER5863 and NSF grant DMS School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, iamen University, iamen, Fujian 365, P.R. China. jxqiu@xmu.edu.cn. Research supported by NSFC grants 93, 384, ISTCP of China Grant No. DFR7.

2 Introduction In this paper, we are interested in solving the hyperbolic conservation law { ut + f(u) x =, u(x, ) = u (x). (.) and its two-dimensional version { ut + f(u) x + g(u) y =, u(x, y, ) = u (x, y), (.) using the Runge-Kutta discontinuous Galerkin methods (RKDG) [7, 6, 5, 8], where u, f(u) and g(u) can be either scalars or vectors. One of the major difficulties to solve (.) and (.) is that the solution may contain discontinuities even if the initial conditions are smooth. Discontinuous Galerkin (DG) methods can capture the weak shocks and other discontinuities without further modification. However, for problems with strong discontinuous solutions, the numerical solution has significant oscillations near discontinuities, which may lead to nonlinear instability. A common strategy to control these spurious oscillations is to apply a nonlinear limiter. Many limiters have been studied in the literature on RKDG methods, including the minmod type total variation bounded (TVB) limiter [7, 6, 5, 8], which is a slope limiter using a technique borrowed from the finite volume methodology [5], and the moment based limiter [] and an improved moment limiter [3], which are specifically designed for discontinuous Galerkin methods to work on the moments of the numerical solution. These limiters do control the oscillations well, however they may degrade accuracy when mistakenly used in smooth regions of the solution. It is usually difficult to design limiters that can achieve both high order accuracy and non-oscillatory properties. Such an attempt has been made in [,, 3, 33, 34, 35, 36, 9], where the weighted essentially non-oscillatory (WENO) finite volume methodology [, 4, 5, 8] or the Hermite WENO finite volume methodology [, 3, 33] are used as limiters for the RKDG methods. More recently, a particularly simple and compact WENO limiter, which utilizes fully the advantage of DG schemes in that a complete polynomial is available in each cell without the need of reconstruction, is designed

3 for RKDG schemes in [3]. The two major advantages of this simple WENO limiter are the compactness of its stencil, which contains only immediate neighboring cells, and the simplicity in implementation, especially for the unstructured meshes [37]. However, it was observed in [3] that the limiter might not be robust enough for problems containing very strong shocks, for example the blast wave problems, making it necessary to apply additional positivity-preserving limiters [3] in such situation. In order to overcome this difficulty, without compromising the advantages of compact stencil and simplicity of linear weights, we present a modification of the limiter in the step of preprocessing the polynomials in the immediate neighboring cells before applying the WENO reconstruction procedure. This preprocessing is necessary to maintain strict conservation, and is designed in [3] to be a simple addition of a constant to make the cell average of the preprocessed neighboring cell polynomial in the target cell matching the original cell average. In this paper, a more involved least square process is used in this step. The objective is to achieve strict conservation while maintaining more information of the original neighboring cell polynomial before applying the WENO procedure. Numerical experiments indicate that this modification does improve the robustness of the limiter. This paper is organized as follows: In Section, we provide a brief review of RKDG for one dimensional and two dimensional cases. In Section 3, we provide the details of the new HWENO limiter for one dimensional scalar and system cases. In Section 4, we provide the details of the HWENO limiter for two dimensional scalar and system cases. We demonstrate the performance of our HWENO limiter with one and two dimensional numerical examples including Euler equations of compressible gas dynamics in Section 5. Concluding remarks are given in Section 6. Review of RKDG methods In this subsection, we give a brief review of the RKDG methods for solving one and two dimensional conservation laws. 3

4 One dimensional case. Given a partition of the computational domain consisting of cells I j = [x j, x j+ cell size by x j = x j+ ], j =,, N, denote the cell center by x j = (x j + x j+ ), and the x j. The DG method has its solution as well as the test function space given by V k h = {v(x) : v(x) I j P k (I j )}, where P k (I j ) denotes the set of polynomials of degree at most k defined on I j. The semi-discrete DG method for solving (.) is defined as follows: find the unique function u h (, t) Vh k such that for j =,, N, (u h ) t v dx f(u h )v x dx + ˆf j+ v(x ) ˆf j+ j v(x + ) = (.3) I j I j j holds for all test functions v V k h, where u± j+ the discontinuous solution u h at the interface x j+ = u h (x ±, t) are the left and right limits of j+ and ˆf j+ = ˆf(u, u + ) is a monotone j+ j+ flux for the scalar case and an exact or approximate Riemann solver for the system case. Two dimensional case. Given a partition of the computational domain consisting of rectangular mesh consisting of cells I ij = [x i and j =,, N y with the cell sizes x i+ centers (x i, y j ) = ( (x i+ + x i ), (y j+ x i, x i+ ] [y j = x i, y j+, y j+ ], for i =,, N x y j = y j and cell + y j )). We now give the new test function space W k h = {p : p I ij P k (I ij )} as the polynomial spaces of degree of at most k on the cell I ij. The semi-discrete DG method for solving (.) is defined as follows: find the unique function u Wh k such that, for all i N x and j N y, (u h ) t v dx dy = I i,j f(u h )v x dx dy I i,j + g(u h )v y dx dy I i,j I j ˆfi+ (y)v(x i+, y) dy + I i ĝ j+ (x)v(x, y j+ ) dx + ˆfi (y)v(x +, y) dy i I j ĝ i (x)v(x, y + ) dx, j I i (.4) hold for all the test function v Wh k, where the hat terms are again numerical fluxes. The semi-discrete schemes (.3) and (.4) can be written as u t = L(u) where L(u) is the spatial discretization operator. They can be discretized in time by a 4

5 non-linearly stable Runge-Kutta time discretization [8], e.g. the third-order version: u () = u n + tl(u n ), u () = 3 4 un + 4 u() + 4 tl(u() ), (.5) u n+ = 3 un + 3 u() + 3 tl(u() ). As in [3], to apply a nonlinear limiter for the RKDG methods, for simplicity, we take the forward Euler time discretization of the semi-discrete scheme (.3) as an example. Starting from a solution u n h V k h at time level n (u h is taken as the L projection of the analytical initial condition into V k h ). We would like to limit it to obtain a new function un,new before advancing it to next time level. That is: find u n+ h Ij u n+ h u n,new h v dx t f(u n,new h I j )v x dx + Vh k, such that, for j =,, N, n,new ˆf v(x ) j+ j+ n,new ˆf v(x + ) = (.6) j j holds for all test functions v V k h. The limiting procedure to go from un h to un,new h will be discussed in the following sections. 3 New HWENO limiter in one dimension In this section, we describe the details of the modified HWENO reconstruction procedure as a limiter for the RKDG method in the one dimensional scalar and system cases. 3. The troubled cell indicator in one dimension An important component of the simple WENO limiter in [3] is the identification of troubled cells, which are cells that may contain discontinuities and in which the WENO limiter is applied. We will use the KRCF shock detection technique developed in [6] to detect troubled cells. We divide the boundary of the target cell I j into two parts: I + j and I j, where the flow is into (v n <, n is the normal vector to I j and v is the velocity) and out of (v n > ) I j, respectively. Here, we define v, taking its value from inside the cell I j as f (u) and take u as the indicator variable for the scalar case. The target cell I j is 5

6 identified as a troubled cell when (u I h (x, t) Ij u h (x, t) Il )ds j > C x k+ k, (3.) j I j u h(x, t) Ij where C k is a constant, usually, we take C k = as in [6]. Here I l, for l = j or j +, denotes the neighboring cell sharing the end point I j with I j. u h is the numerical solution corresponding to the indicator variable(s) and u h (x, t) Ij is the standard L norm in the cell I j. 3. HWENO limiting procedure in one dimension: scalar case In this subsection, we present the details of the HWENO limiting procedure for the scalar case. The idea of this new HWENO limiter is to reconstruct a new polynomial on the troubled cell I j which is a convex combination of three polynomials: DG solution polynomial on this cell and the modified DG solution polynomials on its two immediate neighboring cells. The modification procedure is in a least square fashion. The construction of the nonlinear weights in the convex combination coefficients follows the classical WENO procedure. Assume I j is a troubled cell. Step.. Denote the DG solution polynomials of u h on I j, I j+, I j as polynomials p (x), p (x) and p (x), respectively. Now we want to find the modified version of p (x), denoted as p (x) on the cell I j in a least square fashion. The modification procedure is defined as follows: p (x) is the solution of the minimization problem { subject to φ = p, where min φ(x) P k (I j ) φ = x j I j (φ(x) p (x)) dx I j φ(x) dx, p = x j } I j p (x) dx. Here and below denotes the cell average of the function on the target cell., (3.) Similarly, p (x) is the solution of the minimization problem { } min (φ(x) p (x)) dx φ(x) P k (I j+ ) I j+ (3.3) 6

7 subject to φ = p. For notational consistency we denote p (x) = p (x). The final nonlinear HWENO reconstruction polynomial p new (x) is now defined by a convex combination of these modified polynomials: p new (x) = ω p (x) + ω p (x) + ω p (x). (3.4) According to the modification procedure, it is easy to prove that p new has the same cell average and order of accuracy as p if the weights satisfy ω + ω + ω =. The convex combination coefficients ω l, l =,, follow the classical WENO procedure [5, 4, ]. We discuss it in the following steps. Step.. We choose the linear weights denoted by γ, γ, γ. As in [3], we have used complete information of the three polynomials p (x), p (x), p (x) in the three cells I j, I j+, I j, hence we do not have extra requirements on the linear weights in order to maintain the original high order accuracy. The linear weights can be chosen to be any set of positive numbers adding up to one. The choice of these linear weights is then solely based on the consideration of a balance between accuracy and ability to achieve essentially nonoscillatory shock transitions. In all of our numerical tests, following the practice in [, 3], we take γ =.998 and γ = γ =.. Step.3. We compute the smoothness indicators, denoted by β l, l =,,, which measure how smooth the functions p l (x), l =,,, are on the target cell I j. The smaller these smoothness indicators are, the smoother the functions are on the target cell. We use the similar recipe for the smoothness indicators as in [5]: k ( ) β l = x l d l j l! dx p l(x) dx, l =,,. (3.5) l l= I j Step.4. We compute the non-linear weights based on the smoothness indicators: ω i = ω i l= ω, ω l = l γ l (ε + β l ). (3.6) Here ε is a small positive number to avoid the denominator to become zero. We take ε = 6 in our computation. 7

8 Step.5. The final nonlinear HWENO reconstruction polynomial is given by (3.4), i.e. u new h Ij = p new (x) = ω p (x) + ω p (x) + ω p (x). It is easy to verify that p new (x) has the same cell average and order of accuracy as the original one p (x), on the condition that l= ω l =. 3.3 HWENO limiting procedure in one dimension: system case In this subsection, we present the details of the HWENO limiting procedure for one dimensional systems. Consider equation (.) where u and f(u) are vectors with m components. In order to achieve better non-oscillatory qualities, the HWENO reconstruction limiter is used with a local characteristic field decomposition. In this paper, we consider the following Euler system with m = 3. t ρ ρµ E + x ρµ ρµ + p µ(e + p) =, (3.7) where ρ is the density, µ is the x-direction velocity, E is the total energy, p is the pressure and γ =.4 in our test cases. We denote the Jacobian matrix as f (u), where u = (ρ, ρµ, E) T. We then give the left and right eigenvector matrices of such Jacobian matrix as: and L j (u) = B + µ/c B µ + /c B B B µ B B µ/c R j (u) = B µ /c µ c µ µ + c H cµ µ / H + cµ where c = γp/ρ, B = (γ )/c, B = B µ / and H = (E + p)/ρ. B, (3.8), (3.9) Assume the troubled cell I j is detected by the KRCF technique [6] by using (3.), where v = µ is the velocity and again v takes its value from inside the cell I j. We take both the density ρ and the total energy E as the indicator variables. Denote p, p, p to be the 8

9 DG polynomial vectors, corresponding to I j s two immediate neighbors and itself. Then we perform the characteristic-wise HWENO limiting procedure as follows: Step. Compute L j = L j (ū j ) and R j = R j (ū j ) as defined in (3.8) and (3.9), where ū j is the cell average of u on the cell I j. Step.. Project the polynomial vectors p, p, p into the characteristic fields p l = L j p l, l =,,, each of them being a 3-component vector and each component of the vector is a k-th degree polynomial. Step.3. Perform Step. to Step.5 of the HWENO limiting procedure that has been specified for the scalar case, to obtain a new 3-component vector on the troubled cell I j as p new. Step..4. Project p new back into the physical space to get the reconstruction polynomial, i.e. u new h Ij = R j p new. 4 New HWENO limiter in two dimension 4. The troubled cell indicator in two dimension We use the KRCF shock detection technique developed in [6] to detect troubled cells in two dimensions. We divide the boundary of the target cell I ij into two parts: I + ij and I ij, where the flow is into (v n <, n is the normal vector to I ij) and out of (v n > ) I ij, respectively. Here we define v, taking its value from inside the cell I ij, as the vector (f (u), g (u)) and take u as the indicator variable for the scalar case. For the Euler system (4.4), v, again taking its value from inside the cell I ij, is (µ, ν) where µ is the x-direction velocity and ν is the y-direction velocity, and we take both the density ρ and the total energy E as the indicator variables. The target cell I ij is identified as a troubled cell when I ij (u h (x, y, t) Iij u h (x, y, t) Il )ds h k+ ij I ij u h(x, y, t) Iij > C k, (4.) where C k is a constant, usually, we take C k = as in [6]. Here we choose h ij as the radius of the circumscribed circle in I ij, and I l, l = (i, j ); (i, j); (i +, j); (i, j + ), denote 9

10 the neighboring cells sharing the edge(s) in I ij. u h is the numerical solution corresponding to the indicator variable(s) and u h (x, y, t) Iij is the standard L norm in the cell I ij. 4. HWENO limiting procedure in two dimensions: scalar case In this subsection, we give details of the HWENO limiter for the two dimensional scalar case. The idea is similar to the one dimensional case, i.e. to reconstruct a new polynomial on the troubled cell I ij which is a convex combination of the following polynomials: DG solution polynomial on this cell and the modified DG solution polynomials on its immediate neighboring cells. The nonlinear weights in the convex combination coefficients follow the classical WENO procedure. To achieve better non-oscillatory property, the modification procedure now is in a least square fashion with necessary adjustment. Assume I ij is a troubled cell. Step 3.. We select the HWENO reconstruction stencil as S = {I i,j, I i,j, I i+,j, I i,j+, I ij }, for simplicity, we renumber these cells as I l, l =,, 4, and denote the DG solutions on these five cells to be p l (x, y), respectively. Now we want to find the modified version of p l (x, y), denoted as p l (x, y), in a least square fashion with necessary adjustment. The modification procedure not only use the complete information of the polynomial p l (x, y), but also use the cell averages of the polynomials from its other neighbors. For p (x, y), the modification procedure is defined as follows: p (x, y) is the solution of the minimization problem { ( ) ( ) } min (φ(x, y) p (x, y)) dxdy + l L (φ(x, y) p l (x, y))dxdy, φ(x,y) P k (I ) I I l subject to φ = p 4 and L = {,, 3} {l : p l p 4 < max ( p p 4, p p 4, p 3 p 4 )}, where p l = p l (x, y)dxdy is the cell average of the polynomial p l (x, y) on the cell I l I l I l

11 and I l is the area of I l. Here and below denotes the cell average of the function on its own associated cell. For this modification, we try to find the polynomial p, which has the same cell average as the polynomial on the troubled cell p 4, to optimize the distance to p (x, y) and to the cell averages of those useful polynomials on the other neighboring cells. By comparing the distance between the cell averages of the polynomials on the other neighboring cells and the cell average of p 4 on the target cell, if one is not the farthest, then this polynomial is considered useful. Remark: The one dimensional algorithm is consistent with the two dimensional one, because in one dimension we only have two immediate neighbors and hence L = is the empty set. In the extreme case (for example, if p p 4 = p p 4 = p 3 p 4 ), this two dimensional algorithm could degenerate to the one dimensional case and L could also be. Similarly, p (x, y) is the solution of the minimization problem { ( ) ( ) } min (φ(x, y) p (x, y)) dxdy + l L (φ(x, y) p l (x, y))dxdy, φ(x,y) P k (I ) I I l subject to φ = p 4, where L = {,, 3} {l : p l p 4 < max ( p p 4, p p 4, p 3 p 4 )}. p (x, y) is the solution of the minimization problem { ( ) ( ) } min (φ(x, y) p (x, y)) dxdy + l L (φ(x, y) p l (x, y))dxdy, φ(x,y) P k (I ) I I l subject to φ = p 4, where L = {,, 3} {l : p l p 4 < max ( p p 4, p p 4, p 3 p 4 )}. p 3 (x, y) is the solution of the minimization problem { ( ) ( ) } min (φ(x, y) p 3 (x, y)) dxdy + l L3 (φ(x, y) p l (x, y))dxdy, φ(x,y) P k (I 3 ) I 3 I l

12 subject to φ = p 4, where L 3 = {,, } {l : p l p 4 < max ( p p 4, p p 4, p p 4 )}. We also define p 4 (x, y) = p 4 (x, y). Step 3.. We choose the linear weights denoted by γ, γ, γ, γ 3, γ 4. Similar as in the one dimensional case, we put a larger linear weight on the troubled cell and the neighboring cells get smaller linear weights. In all of our numerical tests, following the practice in [, 3], we take γ 4 =.996 and γ = γ = γ = γ 3 =.. Step 3.3. We compute the smoothness indicators, denoted by β l, l =,..., 4, which measure how smooth the functions p l (x, y), l =,..., 4, are on the target cell I ij. We use the similar recipe for the smoothness indicators as in [5,, 7]: β l = k I ij α α = where α = (α, α ) and α = α + α. I ij ( ) α α! x α y α p l (x, y) dx dy, l =,, 4, (4.) Step 3.4. We compute the nonlinear weights based on the smoothness indicators. Step 3.5. The final nonlinear HWENO reconstruction polynomial p new 4 (x, y) is defined by a convex combination of the (modified) polynomials in the stencil: u new h Ii,j = p new 4 (x, y) = 4 ω l p l (x, y). (4.3) It is easy to verify that p new 4 (x, y) has the same cell average and order of accuracy as the original one p 4 (x, y) on condition that 4 l= ω l =. 4.3 HWENO limiting procedure in two dimensions: system case In this subsection, we present the details of the HWENO limiting procedure for two dimensional systems. Consider equation (.) where u, f(u) and g(u) are vectors with m components. In order to achieve better non-oscillatory qualities, the HWENO reconstruction limiter is used with a l=

13 local characteristic field decomposition. In this paper, we only consider the following Euler system with m = 4. t ρ ρµ ρν E + x ρµ ρµ + p ρµν µ(e + p) + y ρν ρµν ρν + p ν(e + p) = (4.4) where ρ is the density, µ is the x-direction velocity, ν is the y-direction velocity, E is the total energy, p is the pressure and γ =.4 in our test cases. We then give the left and right eigenvector matrices of Jacobian matrices f (u) and g (u) as: L x ij(u) = B + µ/c B µ + /c B ν B ν B B µ B ν B B µ/c B µ /c B ν B, (4.5) µ c µ µ + c Rij(u) x = ν ν ν, (4.6) µ + ν H cµ ν H + cµ L y ij (u) = B + ν/c B µ B ν + /c B µ B B µ B ν B B ν/c B µ B ν /c B, (4.7) R y ij (u) = µ µ µ ν c ν ν + c, (4.8) H cν µ µ + ν H + cν 3

14 where c = γp/ρ, B = (γ )/c, B = B (µ + ν )/ and H = (E + p)/ρ. The troubled cell I ij is detected by the KRCF technique [6] using (4.). Denote p l, l =,, 4, to be the DG polynomial vectors, corresponding to I ij s four immediate neighbors and itself. The HWENO limiting procedure is then performed as follows: Step 4.. We reconstruct the new polynomial vectors p x,new 4 and p y,new 4 by using the characteristic-wise HWENO limiting procedure: Step 4.. Compute L x ij = L x ij(ū ij ), L y ij = Ly ij (ū ij), R x ij = R x ij(ū ij ) and R y ij = Rx ij(ū ij ) as defined in (4.5)-(4.8), where again ū ij is the cell average of u on the cell I ij. Step 4... Project the polynomial vectors p l, l =,, 4, into the characteristic fields p x l = Lx ij p l and p y l = Ly ij p l, l =,, 4, each of them being a 4-component vector and each component of the vector is a k-th degree polynomial. Step Perform Step 3. to Step 3.5 of the HWENO limiting procedure that has been specified for the scalar case, to obtain the new 4-component vectors on the troubled cell I ij such as p x,new 4 and p y,new 4. Step Project p x,new 4 and p y,new 4 into the physical space p x,new 4 = R x ij p x,new 4 and p y,new 4 = R y ij p y,new 4, respectively. Step 4.. The final new 4-component vector on the troubled cell I ij is defined as u new h Iij = 4 + p y,new 4. p x,new 5 Numerical results In this section, we provide numerical results to demonstrate the performance of the HWENO reconstruction limiters for the RKDG methods described in previous sections. We would like to remark that, for some cases with fourth order RKDG methods, the usage of the WENO limiters in [3] might not work well or might even break down, while the new HWENO limiters in this paper could obtain good resolutions. The simple Lax-Friedrichs flux is used in all of our numerical tests. We first test the accuracy of the schemes in one and two dimensional problems. We 4

15 adjust the constant C k in (3.) or (4.) to be a small number. from Example 3. to Example 3.4, for the purpose of artificially generating a larger percentage of troubled cells in order to test accuracy when the HWENO reconstruction procedure is enacted in more cells. The CFL number is set to be.3 for the second order (P ),.8 for the third order (P ) and. for the fourth order (P 3 ) RKDG methods both in one and two dimensions. Example 5.. We solve the following nonlinear scalar Burgers equation: ( ) µ µ t + x =, (5.) with the initial condition µ(x, ) =.5 + sin(πx) and periodic boundary conditions. We compute the solution up to t =.5/π, when the solution is still smooth. The errors and numerical orders of accuracy for the RKDG method with the HWENO limiter comparing with the original RKDG method without a limiter are shown in Table 5.. We can see that the HWENO limiter keeps the designed order of accuracy, even when a large percentage of good cells are artificially identified as troubled cells. Example 5.. We solve the D Euler equations (3.7). The initial conditions are: ρ(x, ) = +. sin(πx), µ(x, ) =. and p(x, ) =. Periodic boundary conditions are applied in this test. The exact solution is ρ(x, t) = +. sin(π(x t)). We compute the solution up to t =. The errors and numerical orders of accuracy of the density for the RKDG method with the HWENO limiter comparing with the original RKDG method without a limiter are shown in Table 5.. Similar to the previous example, we can see that the HWENO limiter again keeps the designed order of accuracy when the mesh size is small enough, even when a large percentage of good cells are artificially identified as troubled cells. Example 5.3. We solve the following nonlinear scalar Burgers equation in two dimensions: ( ) µ µ t + x ( ) µ + y = (5.) with the initial condition µ(x, y, ) =.5+sin(π(x+y)/) and periodic boundary conditions in both directions. We compute the solution up to t =.5/π, when the solution is still 5

16 ( ) µ Table 5.: µ t + =. µ(x, ) =.5 + sin(πx). Periodic boundary condition. t =.5/π. x L and L errors. RKDG with the HWENO limiter compared to RKDG without limiter. DG with HWENO limiter DG without limiter cells L error order L error order L error order L error order.e-.3e-.38e-.4e- 5.6E E E E-.67 P 4.E-3..6E E E E E-3.3.E-4..53E E E E E E-5..69E E-5..68E E-3.83E-.7E-3.8E-.6E E-3.7.E E-3.7 P 4.73E E E E E E E E E E E E E E E E E-4.44E-3.76E-4.35E-3.69E E E E-4.5 P E E E E E E E E E E E E E E E E

17 Table 5.: D-Euler equations: initial data ρ(x, ) = +. sin(πx), µ(x, ) =. and p(x, ) =. Periodic boundary condition. t =.. L and L errors. RKDG with the HWENO limiter compared to RKDG without limiter. DG with HWENO limiter DG without limiter cells L error order L error order L error order L error order.55e- 3.53E- 3.6E-3 8.9E-3.73E E E E-3.63 P 4 3.8E E E E E E E E E E-5.9.E E E-6..3E-5..49E-6..7E E-4 8.E-4.3E E-4.77E E E-5 3..E-4.94 P 4.9E E E E E E E E E E E-8 3..E E E E E E-5.83E-4 5.4E E-5.7E E E E P E E E E E E-9 4..E E E E E E E E E E- 4. 7

18 ( Table 5.3: µ t + ) ( µ + x ) µ y =. µ(x, y, ) =.5 + sin(π(x + y)/). Periodic boundary condition. t =.5/π. L and L errors. RKDG with the HWENO limiter compared to RKDG without limiter. DG with HWENO limiter DG without limiter cells L error order L error order L error order L error order 3.99E- 3.7E- 3.9E- 3.4E- 9.78E-3.3.E E-3..5E-.68 P 4 4.3E E E E E-4. 9.E E E E-4..38E E-4..4E E-3.77E- 5.E-3.8E- 8.6E E E E-. P 4 4.E E-3.77.E E E-5.96.E E-5.96.E E E E E E-3 8.E-.9E-3 8.9E-.5E E E E P E E E E E E E E E E E E smooth. The errors and numerical orders of accuracy for the RKDG method with the HWENO limiter comparing with the original RKDG method without limiter are shown in Table 5.3. We can see that the HWENO limiter keeps the designed order of accuracy, even when a large percentage of good cells are artificially identified as troubled cells. Example 5.4. We solve the Euler equations (4.4). The initial conditions are: ρ(x, y, ) = +. sin(π(x + y)), µ(x, y, ) =.7, ν(x, y, ) =.3, p(x, y, ) =. Periodic boundary conditions are applied in both directions. The exact solution is ρ(x, y, t) = +. sin(π(x + y t)). We compute the solution up to t =. The errors and numerical orders of accuracy of the density for the RKDG method with the HWENO limiter comparing with the original RKDG method without a limiter are shown in Table 5.4. Similar to the previous example, we can see that the HWENO limiter again keeps the designed order of accuracy when the mesh size is small enough, even when a large percentage of good cells are artificially identified 8

19 Table 5.4: D-Euler equations: initial data ρ(x, y, ) = +. sin(π(x + y)), µ(x, y, ) =.7, ν(x, y, ) =.3, and p(x, y, ) =. Periodic boundary condition. t =.. L and L errors. RKDG with the HWENO limiter compared to RKDG without limiter. DG with HWENO limiter DG without limiter cells L error order L error order L error order L error order 4.6E- 7.98E-.55E- 4.44E- 5.6E E E E-3.5 P E E E-4.8.5E E E E E E E-4..89E-5.7.3E E-4 5.7E E-4 5.5E-3.6E E-4.63.E E-4.65 P E E E E E E E E E E E E E E E-5 6.7E-4.85E E E E P E E E E E E E E E E E E-9 4. as troubled cells. We now test the performance of the RKDG method with the HWENO limiters for problems containing shocks. From now on, we reset the constant to C k =. For comparison with the RKDG method using the minmod TVB limiter, we refer to the results in [5, 8]. For comparison with the RKDG method using the previous versions of HWENO type limiters, we refer to the results in [9, 36]. Example 5.5. We consider the D Euler equations (3.7) with a Riemann initial condition for the Lax problem: (ρ, µ, p) T = (.445,.698, 3.58) T for x ; (ρ, µ, p) T = (.5,,.57) T for x >. The computed density ρ is plotted at t =.3 against the exact solution in Figure 5. and the time history of the troubled cells is shown in Figure 5.. We observe that the new RKDG methods do a better job in keeping sharp contact discontinuities and simultaneously saving the computing time. 9

20 Density.8 Density.8 Density Figure 5.: The Lax problem. RKDG with the HWENO limiter. Solid line: the exact solution; squares: numerical solution. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells: T.7.6 T.7.6 T Figure 5.: The Lax problem. RKDG with the HWENO limiter. Troubled cells. Squares denote cells which are identified as troubled cells subject to the HWENO limiting. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells:.

21 Example 5.6. A higher order scheme would show its advantage when the solution contains both shocks and complex smooth region structures. the problem of shock interaction with entropy waves [9]. A typical example for this is We solve the Euler equations (3.7) with a moving Mach = 3 shock interacting with sine waves in density: (ρ, µ, p) T = ( ,.69369, ) T for x < 4; (ρ, µ, p) T = ( +. sin(5x),, ) T for x 4. The computed density ρ is plotted at t =.8 against the referenced exact solution which is a converged solution computed by the fifth order finite difference WENO scheme [5] with grid points in Figure 5.3 and the time history of the troubled cells is shown in Figure 5.4. We observe that the new RKDG methods do well and simultaneously saving the computing time Density 3.5 Density 3.5 Density Figure 5.3: The shock density wave interaction problem. RKDG with the HWENO limiter. Solid line: the exact solution; squares: numerical solution. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells:. Example 5.7. We consider the interaction of blast waves of Euler equation (3.7) with the initial conditions: (ρ, µ, p) T = (,, ) T for x <.; (ρ, µ, p) T = (,,.) T for. x <.9; (ρ, µ, p) T = (,, ) T for.9 x.. The computed density ρ is plotted at t =.38 against the reference exact solution which is a converged solution computed by the fifth order finite difference WENO scheme [5] with grid points in Figure 5.5 and the time history of the troubled cells is shown in Figure 5.6. We observe that the new RKDG methods do a better job than in [3] and simultaneously saving the computing time.

22 T T T Figure 5.4: The shock density wave interaction problem. RKDG with the HWENO limiter. Troubled cells. Squares denote cells which are identified as troubled cells subject to the HWENO limiting. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells: Density 4 3 Density 4 3 Density Figure 5.5: The blast wave problem. RKDG with the HWENO limiter. Solid line: the exact solution; squares: numerical solution. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells: T. T. T Figure 5.6: The blast wave problem. RKDG with the HWENO limiter. Troubled cells. Squares denote cells which are identified as troubled cells subject to the HWENO limiting. Left: second order (P ); middle: third order (P ); right: fourth order (P 3 ). Cells: 4.

23 Remark. For the P 3 case, the RKDG methods with the WENO limiters specified in [3] used the positivity-preserving limiter to avoid negative density or negative pressure during the time evolution, but the resolution was affected, especially for the smearing of contact discontinuities. The RKDG methods with the new HWENO limiters in this paper can work well without the help from the positivity-preserving limiter. Example 5.8. Double Mach reflection problem. This model problem is originally from [3]. We solve the Euler equations (4.4) in a computational domain of [, 4] [, ]. The reflection boundary condition is used at the wall, which for the rest of the bottom boundary (the part from x = to x = ), the exact post-shock condition is imposed. At the top boundary is the 6 exact motion of the Mach shock. The results shown are at t =.. Three different orders of accuracy for the RKDG with HWENO limiters, k=, k= and k=3 (second order, third order and fourth order), are used in the numerical experiments. The simulation results are shown in Figure 5.7. The zoomed-in pictures around the double Mach stem to show more details are given in Figure 5.8. The troubled cells identified at the last time step are shown in Figure 5.9. Clearly, the resolution improves with an increasing k on the same mesh. 6 Concluding remarks We constructed a class of Hermite weighted essentially non-oscillatory (WENO) limiters, based on the procedure of [3], for the Runge-Kutta discontinuous Galerkin (RKDG) methods. The general framework of such HWENO limiters for RKDG methods, namely first identifying troubled cells subject to the HWENO limiting, then reconstructing the polynomial solution inside the troubled cell by the freedoms of the solutions of the DG method on the target cell and its adjacent neighboring cells by a HWENO procedure in a least square sense, is followed in this paper. The main novelty of this paper is the HWENO reconstruction procedure, which uses only the information from the troubled cell and its immediate neighbors (two cells in D and four cells in D), without any other extensive usage of geometric information of the meshes, and with simple positive linear weights in the reconstruction 3

24 Y.5 3 Y.5 3 Y.5 3 Figure 5.7: Double Mach refection problem. RKDG with the HWENO limiter. 3 equally spaced density contours from.5 to.5. Top: second order (P ); middle: third order (P ); bottom: fourth order (P 3 ). Cells: 8. 4

25 .5.5 Y Y Y.5.5 Figure 5.8: Double Mach refection problem. RKDG with HWENO limiter. Zoom-in pictures around the Mach stem. 3 equally spaced density contours from.5 to.5. Left: second order (P ); right: third order (P ); bottom: fourth order (P 3 ). Cells: 8. 5

26 Y.5 3 Y.5 3 Y.5 3 Figure 5.9: Double Mach refection problem. RKDG with the HWENO limiter. Troubled cells. Squares denote cells which are identified as troubled cells subject to the HWENO limiting. Top: second order (P ); middle: third order (P ); bottom: fourth order (P 3 ). Cells: 8. 6

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