Numerical resolution of discontinuous Galerkin methods for time dependent. wave equations 1. Abstract

Transcription

1 Numerical resolution of discontinuous Galerkin methods for time dependent wave equations Xinghui Zhong 2 and Chi-Wang Shu Abstract The discontinuous Galerkin DG method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k+2 superconvergent at Radau points on each element and of order 2k + superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis. Keywords: discontinuous Galerkin method; error analysis; superconvergence; fully discretized; points per wavelength Research supported by NSF grant DMS and DOE grant DE-FG02-08ER Division of Applied Mathematics, Brown University, Providence, RI xzhong@dam.brown.edu Division of Applied Mathematics, Brown University, Providence, RI shu@dam.brown.edu

2 Introduction In this paper, we consider the following time dependent linear wave problem u t + au x = 0, x [0, 2π], t > 0 ux, 0 = e iωx, x [0, 2π]. with periodic boundary conditions, where a is the phase speed for simplicity, assume that a > 0, i = and ω is the wave number for convenience, assume that ω > 0. We develop a quantitative error analysis for the discontinuous Galerkin DG solution via Fourier analysis and study the superconvergence property of the DG solution. The results of this paper provide useful guidelines when the DG method is used to solve more complicated linear and nonlinear wave equations. DG methods are a class of finite element methods using completely discontinuous piecewise polynomials as basis functions. The first DG method was introduced by Reed and Hill [] in 97, to solve the neutron transport equation. The type of DG methods we will discuss in this paper is the Runge-Kutta discontinuous Galerkin RKDG method [22, 2, 9, 8, 2], which is using explicit and nonlinearly stable high order Runge-Kutta method for time discretization and the DG method for space discretization. This method has several advantages, such as local conservation, the allowance of arbitrary triangulation, excellent parallel efficiency, the capability in h-p adaptivity and the ability to sharply capture discontinuities especially contact discontinuities in the solution. Furthermore, it has certain superconvergence properties. The superconvergence behavior of the classical finite element method has been analyzed for many years. For example, in [25], Douglas and Dupont showed that for a general class of two-point boundary value problems, the rate of convergence at the mesh points is of order 2k when polynomials of degree k are used. In [0], Bakker proved that the C 0 Galerkin solution of a two-point boundary value problem using piecewise P k polynomials, has superconvergence of order 2k at the knots and of order k + 2 at the Lobatto points of each segment, and the gradient is superconvergent of order k + at the zeros of a Legendre polynomial shifted to 2

3 the elements of the partition. In [], the same author proved that for two classes of Galerkin methods: the Ritz-Galerkin method for 2m-th order self-adjoint boundary value problems and the collocation method for arbitrary m-th order boundary value problems, the solution has the superconvergence of order k + 2 at the zeros of the Jacobi polynomial Pn m,m σ shifted to the elements of the partition, and the derivative of the solution is superconvergent of order k + at the zeros of the Jacobi polynomial P m,m n+ σ shifted to the elements, where n = k + 2m and k is the degree of the finite element space. Based on the results of Adjerid et al. [] for singularly-perturbed parabolic systems, Biswas et al. [2] used the assumption that the DG solution of hyperbolic conservation laws using k-degree polynomial approximation exhibited superconvergence at the roots of Radau polynomial of degree k + Radau points, to construct a posteriori estimate of spatial discretization error. Adjerid et al. [2] proved that the DG solution of the ordinary differential equation ODE u fu = 0 is superconvergent of order 2k + at the downwind end of each element while maintaining an order of k + 2 at the remaining Radau points. Numerical examples for the partial differential equations PDEs were also shown without analysis. Later, these results were extended to two-dimensional problems on rectangular meshes [] and nonlinear hyperbolic problems [5]. Castillo [] investigated the existence of superconvergent points for DG methods applied to elliptic problems and showed that on each element the k-degree local discontinuous Galerkin LDG solution gradient is superconvergent of order k + at the shifted roots of the k-degree Legendre polynomial. For these cases, the model problems are independent of time. In [], Adjerid et al. showed that the LDG solutions of convection-dominated problems are superconvergent of order k +2 at the shifted Radau points on each element. For diffusiondominated problems, the derivative of the LDG solution is superconvergent of order k + 2 at Radau points. Later, Adjerid et al. proved that the DG solution is superconvergent of order k + 2 at Radau points for linear symmetric hyperbolic systems [6] and for linear

4 symmetrizable hyperbolic systems [7], when t = O. In this paper, by investigating the quantitative error at Radau points using Fourier analysis, we show that, even for t greater than O, the errors at Radau points are of order k + 2 with the exception of the Radau point at the downwind end of the element, which is of order 2k +, when using DG method with piecewise polynomials of degree k. See, for example [29,, 5, 6, 7, 20] for more superconvergence results of the DG method. This work was motivated by [0], where the error analysis is given using the finite difference method. We first provide an overview of this method solving the linear wave problem.. The exact solution of. is Ux, t = e iωx at. Assume a uniform grid with x = 2π, using the following second order central finite difference N method to approximate the spatial derivative in. D 0 u = u j+ u j, 2 x yields a semi-discrete version of. with a system of differential-difference equations given by du j t dt + a u j+t u j t 2 x It is easy to compute the solution of.2, which is = 0, u j 0 = e iωx j..2 u j t = e iωx j ct with the numerical phase speed c = a sin ξ, ξ = ω x. ξ By Taylor expansion, the leading term in the error between the exact solution Ux, t and the approximation solution ux, t is e = U u := max Ux j, t u j t ωatξ2.. 0 j N 6

5 It is evident that the error is a function of time. Under the assumption of periodicity, the important quantity is not the time elapsed, but rather the number of time periods, which is denoted by q, that is, q = t, t = ωat.. 2π Now the critical issue is: how many gridpoints are needed to resolve a wave, namely to make sure that the error is smaller than a given tolerance ε? We introduce the number of points per wavelength, M, which is given by M = N ω = 2π ξ..5 Note that M has a theoretical minimum of 2, since it takes a minimum of two points per wavelength to uniquely specify a wave. Rewriting the error. in terms of M and q, yields Then, the lower bound on M e πq 2 2π..6 M π /2 q /2 M 2π, ε required to ensure a specific error ε can be obtained from.6. According to the analysis above, assume that we know for a given problem the maximum wave number ω needed to adequately describe the solution. Then we can choose the number of points such that the wave is well resolved. This can reduce the cost of computation while preserving the accuracy. If more detailed information is known about the spectral distribution of the Fourier coefficients, then this can be used to obtain sharper comparisons of efficiency by weighting the error function appropriately. In 97, Swartz and Wendroff [] extended the ideas of [0] to the finite element method, using smooth splines as basis functions and presented an analysis of fully discretized approximation. Our approach is to combine and extend the ideas of [0] and [] to the DG method, 5

6 using the techniques introduced in [5] see also [, 6] where the explicit formula of the DG solution is given by Fourier analysis. See, for example [2, 27, 28, 8, 9] for recent development and studies of the dispersion and dissipation errors of DG methods using Fourier analysis. This paper is organized as follows: we first present the DG algorithm formulation for the model problem in Section 2. In Section, we present the details of the error analysis by Fourier analysis and provide the estimates of the necessary number of points per wavelength required to obtain a fixed error. In Section, we show that the DG solution is superconvergent at Radau points. In Section 5, an analysis of fully discretized approximation with Runge-Kutta methods as time discretization is presented. In Section 6, numerical experiments are provided to validate the results predicted by the analysis. Finally, concluding remarks are provided in Section 7. 2 Discontinuous Galerkin formulation In this section, we present the algorithm formulation of the DG method solving the model problem u t + au x = 0, x [0, 2π], t > 0 ux, 0 = sin ωx, x [0, 2π] with periodic boundary conditions. The exact solution to 2. is 2. Ux, t = sinωx at. 2.2 To define the DG method for the model problem, we consider a uniform partition of the computational domain [0, 2π] in N cells of size x = 2π. Denote the cell by I N j = [x j 2 and the cell center by x j = x 2 j+ + x 2 j, j =,, N, where 2 0 = x 2 and denote the approximation space as < x 2 < < x N+ 2 = 2π, x j+ ] 2 V k h = { v : v Ij P k I j ; j N } 2. 6

7 where P k I j denotes the set of polynomials of degree up to k defined on the cell I j. The semi-discrete DG method using the upwind flux for solving 2. is defined as follows: find the unique function u = ut V k h such that, for j =,, N, u t v dx a uv x dx + au j+ I j I j 2vx au j+ 2 j 2vx + = 0 2. j 2 holds for all test functions v V k h. Here and below u+, u denote the left and right limits of the function u at the cell interface, respectively. We now look at the implementation of the scheme 2.. If a local basis of P k I j is chosen and denoted as φ l jx for l =, 2,, k +, then the numerical solution can be represented as k+ ux = u l j φl j x, x I j. 2.5 l= After substituting 2.5 into 2. and inverting a local k + k + mass matrix, the DG scheme 2. can be written as du j dt = a x Au j + Bu j, 2.6 where u j = u j,, T, uk+ j A and B are k + k + constant matrices. As in [5], the local basis of P k I j is chosen to be the Lagrangian polynomials, based on the following k + equally spaced points x j+ 2l k 2k+ = x j + 2l k x, l = 0,, k. 2k + In this way, u j, the coefficients of the solution u inside the cell I j is a vector of length k + containing the values of the solution at these points. The DG finite element scheme 2.6 becomes a finite difference scheme on a globally uniform mesh with a mesh size x k+. However, it is not a standard finite difference scheme because each point in the group of k + points belonging to the cell I j obeys a different form. 7

8 Error analysis In this section, we present the details of DG algorithm formulation with the basis functions discussed in Section 2 and derive the error estimation of the semi-discrete scheme using piecewise P k polynomials, with k =, 2,. At the end of this section, We provide an estimate of the necessary number of points per wavelength required to obtain a fixed error.. The case of P In this subsection, we consider the piecewise linear case, i.e. k =. We present the details of the error analysis for this case. In this case, the local basis functions inside cell I j are φ j x, φ j+ x, which are La- grangian polynomials based on the points x j, x j+. With these basis functions, the solu- tion inside the cell I j is then represented by ux = u j φ j x + u j+ φ j+ x. The finite difference representation of the DG method is u = a 5u j x j 5 + 5u j 7u j u j+ = a x + u j+ u j 5 u j + u j 9u j+ The scheme. can be rewritten into the matrix form.. du j dt = a x Au j + Bu j,.2 where u j = uj u j+, A = 7 9, B = To solve.2, the standard Fourier analysis is used here. This analysis depends heavily on the assumption of uniform mesh and periodic boundary conditions. Assume uj t û t = t t u j+ 8 û e iωx j,.

9 and after substituting. into the DG scheme.2, the coefficient vector satisfies the following ODE system û û t t = G û t û t,.5 where G is the amplification matrix, given by The two eigenvalues of G are G = a x A + Be iξ, ξ = ω x..6 λ,2 = a e iξ 2 e x 2iξ + 0e iξ 2,.7 and the corresponding eigenvectors are + e iξ + 0e iξ 2e 2iξ V,2 = + e iξ..8 Then the general solution of the ODE system.5 is û t û t = C e λ t V + C 2 e λ 2t V 2..9 The coefficients C and C 2 in.9 can be determined by the initial condition i ξ û 0 û 0 = e e i ξ..0 We thus have the explicit solution of the DG scheme with piecewise linear polynomials for solving 2.. Comparing with the exact solution 2.2 will give us the quantitative error. Now let us first introduce some notations: e := max j N e + := max j N Ux j Ux j+, t u j, t u j+ t, t.. 9

10 By a simple Taylor expansion, we get e + = 2 ξ2 + t 72 ξ ξ 86 t 720 ξ t 2 + ξ t t ξ t t t t t ξ 8 ξ Oξ 0 where t = ωat. From.2, for short time t, the first term 2 ξ2, which is independent of t, dominates. As t increases to O, the second term t ξ 72 ξ begins to dominate. If t continues increasing, another term with higher degree of t as coefficient may dominate. This means, during different time intervals, the dominant terms are different. At t = O, the error can be approximated by the first two terms, i.e. e 2 ξ2 72 tξ, e + 2 ξ tξ. ξ. Denote e = max { } e, e +, where the subscript is used here to indicate the case k =. Clearly, the error e satisfies e = max { } e, e + 2 ξ tξ.. Similar to the finite difference case, we would like to rewrite the error in terms of the number of points per wavelength M and the number of time periods q = t. Notice that two points 2π 0

11 {x j, x j+ } are used for each cell, therefore, M = 2N ω = π ξ..5 Substituting.5 and. into., we get e 2 π + πq 2 M 6 π..6 Then the number of points per wavelength required to guarantee the error e ε satisfies the following inequality.2 The case of P 2 M 2 π + πq π ε..7 2 M 6 M In this subsection, we use the same procedure as we did in the piecewise linear case to present the details of the error estimation of the DG method using piecewise P 2 polynomials. In this case, the local basis functions { φ j x, φ j x, φ j+ x } inside cell I j are the Lagrangian polynomials based on the points x j, x j, x j+. The solution inside the cell I j is then represented by ux = u j φ j x + u j φ j x + u j+ φ j+ x, and the DG scheme can be written into the matrix form 2.6 with u j u j = u j, A = , B = u j The standard Fourier analysis is again used here. First we make an ansatz of the form u j t u j t u j+ t = û t û 0 t û t eiωx j..9

12 After substituting.9 into the DG scheme 2.6 with.8, the coefficient vector satisfies û û t t û 0 t = G û 0 t,.20 û t û t where G is given by.6 with A and B defined in.8. Then with three eigenvalues of G and the corresponding eigenvectors computed by Mathematica, the general solution of the ODE system.20 is û t û 0 t û t = C 2 e λ t V + C 22 e λ 2t V 2 + C 2 e λ t V..2 The coefficients C 2, C 22 and C 2 in.2 can be determined by the initial condition û 0 e iξ û 0 t û 0 = e iξ..22 We thus have the explicit solution of the DG scheme with P 2 polynomials for solving 2.. Comparing with the exact solution 2.2 will give us the quantitative error estimates. e := max Ux j, t u j N j t = ξ ξ5 2 tξ t2 ξ 7 + Oξ 8, e 0 := max j N Ux j, t u j t e + := max j N = ξ ξ5 tξ Ux j+, t u j+ t = 2ξ ξ5 26 tξ t ξ 7 + Oξ 8, t ξ 7 + Oξ 8. Clearly, the coefficients of high order terms depend on t. Similarly, during different time intervals, the dominant terms are different. Denote e 2 := max { } e, e 0, e +. 2

13 At t = O ξ, the error e 2 satisfies At t = O, the error e ξ 2 2 satisfies e 2 ξ e 2 ξ ξ5 tξ t2 ξ In this case, the number of points per wavelength M 2 satisfies M 2 = N ω = 6π ξ,.25 since three points are used in each cell. Similarly, by rewriting.2 and.2 in terms of M 2 and q = t 2π, we can obtain the number of points per wavelength M 2 required to ensure that e 2 ε.. The case of P In this subsection, we apply the same procedure discussed in the previous subsections to the DG method using piecewise P polynomials. { The local basis functions φ j x, φ 8 j 8 Lagrangian polynomials based on the points x j 8 x, φ j+ 8, x j 8 } x, φ j+ x 8, x j+ 8, x j+. With these basis func- 8 tions, the DG scheme can be written into the matrix form 2.6 with A = inside cell I j are the

14 and B = The standard Fourier analysis is one again used here to obtain the explicit solution of the DG scheme. The error is also approximated by the dominant terms of Taylor series. Denote { e := max e, e 8, e 8 +, e 8 + }. At t = O, 8 ξ 2 At t = O, ξ. e ξ..26 e ξ ξ tξ 7. In this case, the number of points per wavelength M satisfies M = N ω = 8π ξ,.28 since four points are used in each cell. Similarly, the number of points per wavelength M required to ensure that e ε can be obtained by rewriting.26 and.27 in terms of M and q. Table. shows the lower bound of M k to ensure the specific error ε, which is a function of q. we only present the leading term here. Superconvergence In this section, we study the superconvergence property of the DG solution at Radau points.

15 Table.: Leading term of the lower bound of M k as a function of q M M 2 M t O O O O O ξ ξ ξ 2 ξ 2 ε = 0. 2q 6.5 7q q 7 ε = q.08 0q q 7 ε = q 0. q q 7 ξ. Superconvergence at Radau points In this subsection, we study the superconvergence property of DG solutions at Radau points. The local basis functions of P k I j are chosen to be the Lagrangian polynomials, based on the following k + points x j+rl = x j + ζ k,l x, l = 0,, k 2 where {ζ k,l } are the roots of the Radau polynomial P k+ x P k x and P k x is the Legendre polynomial of degree k. Table. displays {ζ k,l } for k =, 2,. Note that x j+rk = x j+ 2 the downwind point of the element I j. is Table.: {ζ k,l } k l=0 : the roots of P k+x P k x ζ k,l k = ζ,0 =, ζ, = k = 2 ζ 2,0 = + 6, ζ 5 2, = 6, ζ 5 2,2 = ζ,0 = ζ k =, = ζ,2 = ζ, = Remark: The basis functions in this section are different from those discussed in Section and the amplification matrices are different, but they have the same eigenvalues. These eigenvalues are also the same with the ones derived in [2] using an orthonormal basis of Legendre polynomials. As in Section 2, the DG method with the above basis can be written into a matrix form. The standard Fourier analysis is used to find the explicit solution of the DG scheme as 5

16 discussed in Section. We do not repeat the details of this analysis procedure. Denote e rl := max Ux j+rl, t ux j+rl, t, j N e + 2 := max j N The followings are the error estimate results. Ux, t ux, t j+ j k = e r0 = t2 ξ + Oξ, e + = t2 ξ + Oξ ; k = 2 e r0 = ξ t 7200 ξ5 + Oξ6, e r = ξ + t 7200 ξ5 + Oξ 6, t 2 + /25 e + = ξ 5 + Oξ 6 ; k = e r0 = ξ ξ tξ 8 + Oξ 9, e r = ξ ξ tξ 8 + Oξ 9, e r2 = ξ ξ tξ 8 + Oξ 9, 6

17 t /9 e + = ξ t t 2 ξ 8 + Oξ 9..2 Superconvergence at the downwind point In this subsection, we take a particular look at the error at the downwind point of each cell. Let M f k be the number of points per wavelength needed to guarantee the error e + ε 2 when using P k polynomials. According to the results derived in the previous subsection, when t is greater than O, we have k = e + = t2 ξ + Oξ t 72 ξ = πq π ε.. 6 M f k = 2 e + 2 = t 2 + /25 ξ 5 + Oξ t 7200 ξ5 = πq 5 6π ε M f 2 k = e + 2 = t /9 ξ 7 + Oξ t 200 ξ7 = πq 7 8π ε M f Therefore, π M f π q 6 ε π M f 2 6π 5 q 600 ε M f 8π π ,. 5 7 q ε, Table.2 shows the lower bound of M f k as a function of q for three representative values of ε, when t is greater than O. 7

18 Table.2: Lower bound of M f p as a function of q 5 Fully Discrete Schemes M f M f 2 M f ε = 0. 2q 7q 5 6q 7 ε = q q 5 8q 7 ε = q 8q 5 q 7 The previous sections deal with the semi-discrete schemes. In practice, the time discretization also plays an important role. An analysis of fully discretized approximation will be presented in this section with the basis functions discussed in Section. 5. The case of P In this subsection, we present the details of fully discretized error analysis for the piecewise linear case. Let ũ be the approximation solution of the differential-difference equations 2. obtained by using p-stage explicit Runge-Kutta methods of order p RKp, p [26]. Then ũj = R n eiωx j, n = t ũ j+ e iωx j+ t 5. where R = + tg + t2 2! G2 + + tp p! Gp 5.2 with the amplification matrix G defined in.6. Let Q = [V, V 2 ] be the matrix with G s eigenvectors as columns. Clearly, we have Q λ 0 GQ = Λ = 0 λ Denote R = Q RQ. From 5. and 5.2, we have R = Q RQ = + tλ + + tp + tλ + tp λ p p! Λp p! 0 = 0 + tλ 2 + tp λ p p! 2. 8

19 Thus, R n = Q RQ n = Q R + n Q tλ + tp λ p p! 0 = Q 0 + tλ 2 + tp λ p p! 2 n Q. 5. Substituting 5. into 5., we can determine the explicit solution of the RKDG scheme for solving 2.. Comparing with the exact solution 2.2 will give us the quantitative error estimates, based on the assumption cfl = a t x. By a simple Taylor expansion, the error between the DG solution with RK2,2 time discretization and the exact solution is ũ j Ux j, t = ξ2 icfl 2 t e it ξ2 2 2 where ξ = ω x and t = ωat. + ξ 576 i 8t 2cfl cfl t e it + ξ 576 i 6 cfl + 8 cfl + ξ 2 t 6 cfl + 8 cfl 6 cfl + 8 cfl 2 at i +6 cflt cfl x e 6 cfl+8 cfl 2 at i +6 cflt 2 cfl x e 6 cfl+8 cfl 2 at i +6 cflt 2 cfl x e 6 cfl+8 cfl 2 + In order to let the terms with coefficients at 6 cfl + 8 cfl 2 cfl x go to 0 as x 0, we assume 6 cfl + 8 cfl 2 <, i.e. cfl <. Then ũ j ξ Ux j, t = e it 2 2 ξ2 6 i cfl2 t i ξ 576 ξ 72 t ξ 2 cfl2 t + ξ 8 cfl t + Oξ. Thus e = max j N = 2 ũ j Ux j + 6cfl t 2 ξ 2, t 2t + 7cfl 2 t 8cfl t ξ + 6cfl t 2 + Oξ. 5.5 We keep the same notations here as in the semi-discrete case. Similarly, e + = 2t + 7cfl + 6cfl t 2 2 ξ 2 t 8cfl t ξ + + Oξ cfl t 2 9

20 and e = max{ e, e + } = 2t + 7cfl + 6cfl t 2 2 ξ 2 t 8cfl t ξ + + Oξ 2 + 6cfl t Since it takes a minimum of two points per wavelength to uniquely specify a wave, the largest wave number that can be represented on the N cells reminder: two points are used for each cell is ω = 2N 2 = N. Therefore, we consider simple waves with 0 < ω N. If ω is much less than N in magnitude, then there are many gridpoints per wavelength, and the solution is well resolved. In this case, ξ is a small number, and we can neglect Oξ and obtain e + 6cfl t 2 ξ If ξ is large, corresponding to fewer gridpoints per wavelength in the solution, then P polynomials will not yield a good approximation of x degree > are required. and approximation polynomials of Similarly, when using RK, time discretization, the error estimates under the assumption cfl < 0.8 are At t = O ξ, e = ξ 2 2 ξ 72 t + cfl + Oξ, 5.9 e + = ξ2 2 + ξ 72 t + cfl + Oξ. 5.0 e ξ2 2 + ξ 72 t + cfl. 5. The necessary number of points per wavelength required to ensure a specific error ε can be obtained by rewriting 5.8 and5. in terms of M and q and setting e ε. 5.2 The case of P 2 For k = 2 and k =, the analysis procedure is the same. 20

21 In this subsection, we present the results of fully discretized error estimates when piecewise P 2 polynomials are used. When using RK, time discretization, under the assumption cfl < 0.27, we have e = 296 e 0 = 20 e + = cfl6 t 2 ξ + Oξ cfl 6 t 2 ξ + Oξ, cfl6 t 2 ξ + Oξ. 5. Thus, e 2 = max{ e, e 0, e + } + 00cfl 6 t 2 ξ When using RK, time discretization, under the assumption cfl < 0.5, we have At t = O ξ, e = 296 ξ + t 20 cfl ξ + Oξ 5, 5.6 e 0 = 20 ξ + t 20 cfl ξ + Oξ 5, 5.7 e + = ξ t 20 cfl ξ + Oξ e 2 20 ξ + t 20 cfl ξ. 5.9 While at t = O, we need more terms of the Taylor series to approximate e ξ 2 2. In this 2

22 case, e ξ cfl ξ t ξ 5 + ξ t cfl t cfl 5 t cfl 6 t + ξ t cfl t cfl 5 t cfl 8 t cfl 9 t cfl 0 t 2. The necessary number of points per wavelength required to ensure a specific error ε can be obtained by rewriting 5.5, 5.9 and 5.20 in terms of M 2 and q and setting e 2 ε. 5. The case of P For the cases k = and k = 2, the cfl condition is consistent with the CFL L 2 table in [2]. However, for k =, we cannot get the cfl condition using the same procedure by Taylor expansion and Mathematica due to limited memory. In this subsection, we provide the error estimates based on a fixed cfl = 0.. When using RK, time discretization, we have And e = ξ t 2 + Oξ 5, 8 e = ξ t 2 + Oξ 5, 8 e + = ξ t 2 + Oξ 5, 8 e + = ξ t 2 + Oξ 5. 8 e ξ t

23 When using RK5,5 time discretization, we have e = ξ tξ 5 + Oξ 6, e = ξ tξ 5 + Oξ 6, e + = ξ tξ 5 + Oξ 6, e + = ξ tξ 5 + Oξ At t = O, ξ 2 e ξ tξ The necessary number of points per wavelength required to ensure a specific error ε can be obtained by rewriting 5.2 and 5.26 in terms of M and q and setting e ε. Table 5. shows the M k as a function of q we only present the leading term here, derived from the error estimates in each subsection. These estimates is computed at t = O. Here we fix cfl = 0. for k =, cfl = 0.2 for k = 2 and cfl = 0. for k =. Table 5.: Leading term of the lower bound of M k as a function of q M M 2 M RKp, p RK2,2 RK, RK, RK, RK, RK5,5 ε = 0. 2q 2 2q 5q q 2q 2q 5 ε = 0.0 9q 2 26q q 8q q 2q 5 ε = q 2 57q 2q 0q 8q q 5 ξ So far, in every case, we have expressed e k as a function of M k and q. Now the work per wavelength to obtain the error e k when integrating to time t using RKp, p is given by W k = 2k + M k t t p, which is a product of the number of operations per mesh point per time step 2k +, the number of points per wavelength M k = 2πk+ ω x, the total number of time steps t number of stages per time step p. Using ωat = 2πq and cfl = a t, we obtain x W k = 2k + M k p t t t and the 2

24 6 Numerical Results = 2k + M k p 2πq ωa t 2πq = 2k + M k p ω x cfl = 2k + M k p q 2π cfl ω x = 2k + M k p q M k cfl k + = 2 cfl pqm2 k. In this section, we provide numerical experiments to demonstrate the predicted results presented in the previous sections. To verify the predicted results for the semi-discrete scheme, we adopt SSPRK9,9 [26] to make the temporal error negligible compared to the spatial error. This time discretization method for solving du/dt = Lu, where L is a spatial discretization operator for this SSP method to be ninth order, L needs to be linear is defined as follows: where u i = u i + t Lu i, i =,, 8 7 u 9 = α 9,k u k + α 9,8 u 8 + t L u 8, k=0 α m,0 = , α m,0 = , α m,2 = 0 560, α m, = 5 86, α m, = 720, α m,5 = 20, α m,6 = 260, α m,7 = 0080, α m,8 = Uniform meshes are used in the calculation and the CFL condition cfl = a t x the table in [2]. is chosen from Tables list the error between DG solution and exact solution for fixed t with different cell size N, where N = N ω. The numerical results are computed using P k polynomials and RK9,9 time discretization. The predicted results are computed using the formulas 2

25 derived in Section. The predicted errors agree with the computed results very well. These tables also show that the orders of the errors are different at different final time t. This is because the higher order terms of the error dominate for large t. Table 6.: When using P k polynomials on a uniform mesh of N cells, t = N Numerical results Predicted by analysis k = e order e order 20.69E-0.50E E E E E E E k = 2 e 2 order e 2 order E E E E E E E E k = e order e order E E E E E E E E-0.00 Table 6.2: When using P k polynomials on a uniform mesh of N cells, t = 00 N Numerical results Predicted by analysis k = e order e order E E E E E E E E k = 2 e 2 order e 2 order 20.70E-0.6E E E E E E E k = e order e order 20.5E E E E E E E E

26 Table 6.: When using P k polynomials on a uniform mesh of N cells, t = 000 N Numerical results Predicted by analysis k = e order e order E-0.76E E E E E E E k = 2 e 2 order e 2 order 20.8E E E E E E E E-07.0 k = e order e order E E E E E E E E-0.00 Figures show the time evolution of the L error and e k error when using P k polynomials with three different time discretization methods. One is the same order as the spatial error, another one is one order higher than the spatial error and the last one is the RK9,9 which makes the temporal error negligible comparing with the spatial error. The numerical L error is computed by taking a uniform partition of each element with 20 points. These figures show that the time discretization method needs to be at least one order higher than the DG method, in order to get the same result as the semi-discrete case. The order of the RK method higher than k + makes little difference when the DG method is using P k polynomials. Also, the figures demonstrate that the predicted errors in general agree with the computed results very well. Tables list the numerical errors and their orders at Radau points for different final time t. These tables verify the results derived in Section : the errors at the downwind point of each element are superconvergent of order 2k + and at other Radau points are superconvergent of order k +2 when using piecewise P k polynomials. These tables also show the relation between the error and t: for k =, errors are proportional to t while for k = 2,, 26

27 Numerical L error RK2,2 RK, RK9, time 0.6 RK2,2 Numerical RK2,2 Predicted 0. RK, Numerical RK, Predicted 0.2 RK9,9 Numerical 0.0 Semi-dicrete error The error e time Figure 6.: Time evolution of L error left and e error right for P DG solution with N = 0, cfl = 0.. 7E-05 6E-05 5E-05 E-05 Numerical L error RK, RK, RK9,9 7E-05 6E-05 5E-05 E-05 The error e 2 RK, Numerical RK, Predicted RK, Numerical RK, Predicted RK9,9 Numerical Semi-dicrete error E-05 E-05 2E-05 2E-05 E-05 E time time Figure 6.2: Time evolution of L error left and e 2 error right for P 2 DG solution with N = 80, cfl =

28 .5E-09 Numerical L error.5e-09 The error e.0e E E-09 RK, RK5,5 RK9,9.0E E E-09 RK, Numerical RK, Predicted RK5,5 Numerical RK5,5 Predicted RK9,9 Numerical Semi-dicrete error.5e-09.5e-09.0e time.0e time Figure 6.: Time evolution of L error left and e error right for P DG solution with N = 60, cfl = 0.. the leading terms of the errors do not depend on t, thus the errors are almost the same for t under O. We have also tested the P 2 case at final time t = 500, the error max { e ξ r l } 0 l achieves fifth order. This is because as t increases to 500, the fifth order term dominates. This verifies the fact that during different time intervals, the dominant terms are different. We have also used a non-uniform mesh which is 60% random perturbation of the uniform mesh. For example, the right end of the cell I j is now x j+ 2 x j+, x are taking values under the uniform mesh and r 2 j %r j x, where is a random number from the uniform distribution over the range 0,. Tables list the errors and their orders at Radau points in this case for different final time t, when using piecewise P k polynomials. We can see that the error at Radau points is of order around k + 2, including the downwind point. We have also tested the cases of P 2 and P using % random perturbation of the uniform perturbation. The numerical results still do not show the strong superconvergence of order 2k +. Tables 6.0 and 6. list the necessary number of points per wavelength required to guarantee e k ε for various RKDG schemes and values of ε. Denote RKDGp, k + to be the scheme using the RKp, p method for time discretization and the DG method with piecewise P k polynomials for space discretization. For these schemes with RK9,9 method, 28

29 Table 6.: Error at Radau points using P DG and RK9,9 on a uniform mesh of N cells t = t = 0 t = 00 N error order error order error order 20.75E-0.25E-0.7E E E E e r E E E E E E e E-0.28E-0.8E E E E E E E E E E Table 6.5: Error at Radau points using P 2 DG and RK9,9 on a uniform mesh of N cells t = t = 0 t = 00 N error order error order error order 20.05E-05.2E-05.97E-05 max { e r l } 0 6.7E E E l 80.25E E E E E E E-06.E-06.2E-05 e 0.E E E E E E E E E Table 6.6: Error at Radau points using P DG and RK9,9 on a uniform mesh of N cells t = t = 0 t = 00 N error order error order error order 20.6E-07.6E-07.8E-07 max e r l 0.70E E E l 2 80.E E E E E E E E-09 2.E-08 e 0 2.2E E E E E E E E E

30 Table 6.7: Error at Radau points using P DG and RK9,9 on a random mesh of N cells N t = t = 0 t = 00 error order error order error order 20.82E-0 6.7E-0 6.E E E E-0.08 e r0 80.2E E E E E E E E E E-0 6.8E-0 6.E E E E-0.08 e E E E E E E E E E Table 6.8: Error at Radau points using P 2 DG and RK9,9 on a random mesh of N cells N t = t = 0 t = 00 error order error order error order 20.20E-05.E-05.E E E E e r E E E E E E E E E E-05.9E-05 9.E E E E e E E E E E E E E-.26.0E-0.76 Table 6.9: Error at Radau points using P DG and RK9,9 on a random mesh of N cells N t = t = 0 t = 00 error order error order error order 20.90E-07.6E-07.7E E E E e r0 80.0E E E E- 5..8E E E-2..E-2.7.0E E E-08.5E E E E-09.5 e E E E E E E E E E-.7 0

31 the results of semi-discrete case are used as the predicted results. These two tables also verify our conclusions about the error estimates with different time discretization methods, which states that the RK method needs to be at least one order higher than the DG method, in order to get the same result as the semi-discrete case; and there is little difference among the RK methods of order higher than k + when the DG method is using piecewise P k polynomials. Table 6.0: Necessary number of points per wavelength for e k ε when time period q = RKDGp, k + ε = 0. ε = 0.0 ε = 0.00 Numerical Predicted Numerical Predicted Numerical Predicted RKDG2, RKDG, RKDG9, RKDG, RKDG, RKDG9, RKDG, RKDG5, RKDG9, Table 6.: Necessary number of points per wavelength for e k ε when time period q = 0 RKDGp, k + ε = 0. ε = 0.0 ε = 0.00 Numerical Predicted Numerical Predicted Numerical Predicted RKDG2, RKDG, RKDG9, RKDG, RKDG, RKDG9, RKDG, RKDG5, RKDG9, Tables list the work per wavelength of the scheme RKDGp, k+, w k, required to obtain the error ε when time period q = and q = 0. We should point out that we could have increased the order of the RK method used in the scheme RKDGk + 2, k + without significantly affecting the number of points per wavelength, but the numbers of stages per

32 timestep would then increase considerably and so is the work per wavelength. It is clear that even for short time period, high order methods are the most appropriate choice when accuracy is the primary consideration. When using P k polynomials for the DG method, RKk+, k+ needs less work than the RKk+2, k+2 for the short time period. While for long time period, the better choice is RKk + 2, k + 2. Table 6.2: When time period q = Type of Multiplies Stages W Order k Scheme per Meshpoint per Timestep ε = 0. ε = 0.0 ε = 0.00 RKDG2, ,900 2,000 22,700 RKDG,2 2,000 2,000 72,000 RKDG, 6,00 6,00 29,800 RKDG, 6,800 8,00 7,00 RKDG, 8,000 9,600 0,500 RKDG5, 8 5,800 2,000 8,00 Table 6.: When time period q = 0 Type of Multiplies Stages W Order k Scheme per Meshpoint per Timestep ε = 0. ε = 0.0 ε = 0.00 RKDG2, ,00,995,00 9,89,700 RKDG,2 2 9,900 75,000,972,500 RKDG, 6 8,00 76, ,00 RKDG, 6 5,800 57,800 8,600 RKDG, 8 0,600 96,900 06,00 RKDG5, 8 5 8,00 20,800 8,500 In Figure 6., we compare the necessary number of points per wavelength required to obtain a fixed error 0.0 between two second order accurate methods. One is the DG method with P polynomials and the other is the second order finite difference method. It can be seen clearly that, even for a short time period, the DG method needs much less points per wavelength. Also, the necessary number of points per wavelength of the finite difference scheme grows faster with time than the RKDG method. This is because the leading term of the error for the RKDG method does not depend on t, while the finite difference method does. This conclusion also holds for higher order schemes between the DG method and the finite difference method with the same order. In [], Swartz and Wendroff computed the 2

33 necessary number of intervals per wavelength required to obtain a fixed error for the finite element method using smooth splines as basis functions and various time discretization methods, such as the trapezoidal method and the leap-frog method. With the same order of time discretization method and space discretization method, the RKDG method requires less intervals per wavelength than the finite element method with smooth splines as basis functions in order to obtain a specified error. Moreover, the leading term of the error of this finite element method using smooth splines depends on t. This means the necessary number of interval per wavelength for the finite element method using smooth splines also grows faster with time than the RKDG method nd FD P DG ε= q Figure 6.: Necessary Points per Wave for e ε Remark: All the numerical results and the predicted results take point value collocation as initial conditions in this paper. The usual way of taking initial conditions in a finite element method is via an L 2 projection. However this does not affect the results in our paper. See Figure Conclusion In this paper, by Fourier analysis, we have derived the quantitative error estimates of the DG solutions using piecewise P k polynomials with k, for solving time dependent linear

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