ader schemes for hyperbolic conservation laws with ... - CiteSeerX

European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006 P. Wesseling, E. O˜ nate and J. P´ eriaux (Eds) c TU Delft, The Netherlands, 2006 °

ADER SCHEMES FOR HYPERBOLIC CONSERVATION LAWS WITH REACTIVE TERMS V.A. Titarev and E.F. Toro ∗ University

of Trento, Department of Civil and Environmental Engineering, Via Mesiano, 77, 38050, Trento, ITALY e-mail: [email protected], [email protected]

Key words: Hyperbolic systems, Derivative Riemann problem, piece-wise smooth data, evolved-data Riemann solvers, arbitrary-order schemes Abstract. In this paper we generalizew the semi-analytical method 16 for solving the Derivative Riemann Problem to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for two-dimensional hyperbolic systems which illustrate robustness and high accuracy of the resulting schemes.

1

INTRODUCTION

ADER15 is a recent approach for construction of arbitrary high-order advection schemes for hyperbolic conservation laws with reaction-type source terms. The key part of the method is the solution of the so-called Derivative Riemann problem. The DRP is a generalization of the conventional Riemann problem in that the governing system may contain source terms and the initial conditions may consist of polynomials of arbitrary degree. The semi-analytical solution procedure for the DRP reported in16 provides an approximation to the state variable along the t-axis in the form of a Taylor time expansion by reducing the problem to a sequence of conventional Riemann problems. The leading term of the expansion is computed as the Godunov state of the conventional nonlinear Riemann problem, whereas the evaluation of higher-order terms involves the solution of linearized Riemann problems for spatial derivatives. The corresponding ADER-type methods for homogenous systems have been reported in.3, 8, 11, 15 It is obvious that availability of an approximate-state Riemann solver is crucial for building up the approximate solution to the DRP. For complex nonlinear systems such solvers may become very complicated or simply unavailable. It is therefore desirable to have a simple procedure for calculating the leading term of the state expansion which would not necessarily require a detailed knowledge of the Riemann problem solution. The aim of the present paper is threefold. Firstly, we present a new method to compute the leading term of the Taylor time expansion which does not require a Riemann solver 1

V.A. Titarev and E.F. Toro

for the nonlinear system to be solved. This method combines a non-linear evolution of the initial condition of a conventional Riemann problem and a simple linearization of the Riemann problem giving closed-form solutions. Secondly, we incorporate the new variant of the DRP solver into high order finite volume ADER methods for hyperbolic systems with source terms. Finally, wee present numerical results for the compressible Euler equations and nonlinear shallow water equation with a source term. Comparisons with the state-of-the-art weighted essentially non-oscillatory (WENO) schemes5, 9 are provided. These results illustrate that the ADER approach retains designed order of accuracy for inhomogeneous systems and that the new version of the DRP solver works well. The rest of the paper is organized as follows. In Section 2 we review the current DRP solver as applied to nonlinear systems with reactive-type source terms. In Section 3 we present a new procedure to compute the leading term. In Section 4 we describe the application to the ADER approach. Numerical examples for two-dimensional hyperbolic systems are given in Section 5 and conclusions are drawn in Section 6. 2

The Derivative Riemann Problem

The Derivative Riemann Problem or DRP for a hyperbolic system is the initial-value problem ∂t Q + ∂x F(Q) = S(x, t, Q) ,   QL (x) if x < 0 , Q(x, 0) =  QR (x) if x > 0 .

(1)

where the initial states QL (x), QR (x) are two vectors, the components of which are smooth functions of distance x. We introduce the notation DRPK to mean the Derivative Riemann Problem in which K represents the number of non-trivial spatial derivatives of the initial condition, K = max{KL , KR }, where KL and KR are such that ∂x(k) QL (x) ≡ 0 ∀k > KL , ∀x < 0 and ∂x(k) QR (x) ≡ 0 ∀k > KR , ∀x > 0 . DRP0 means that all first (k = 1) and higher-order spatial derivatives of the initial condition for the DRP away from the origin vanish identically; this case corresponds to the conventional piece-wise constant data Riemann problem. We are interested in the solution of DRPK , right at x = 0, where the initial data may be discontinuous. We seek a power series solution at x = 0 as a function QLR (t) of time t only as K h i τk X (k) , (2) ∂t Q(0, 0+ ) QLR (τ ) = Q(0, 0+ ) + k! k=1 where

0+ ≡ lim t. The solution contains a leading term Q(0, 0+ ) and higher-order t→0+

(k)

terms with coefficients determined by ∂t Q(0, 0+ ). In what follows we describe a method to compute each of the terms of the series expansion. 2

V.A. Titarev and E.F. Toro

The leading term Q(0, 0+ ) in the expansion accounts for the first-instant interaction of the initial data via the governing PDEs, which is realized solely by the boundary extrapolated values QL (0) and QR (0) in (1). Therefore, Q(0, 0+ ) is found from the similarity solution of the following DRP0 ∂t Q + ∂x F(Q) = 0 , Q(x, 0) =

  QL (0) ≡ limx→0 QL (x) −

if

x0.

(3)

Here, the influence of the source term can be neglected. Denoting the similarity solution by D(0) (x/t), the sought leading term is given by evaluating this solution along the t-axis, that is along x/t = 0, namely Q(0, 0+ ) = D(0) (0) .

(4)

The value D(0) (0) is commonly known as the Godunov state, as it is used in the numerical flux of the first-order upwind scheme of Godunov.4 In what follows we shall extend the use of this terminology to mean the solution of conventional Riemann problems for spatial derivatives evaluated at x/t = 0. In practice, a conventional Riemann solver, possibly approximate, is needed here to determine the leading term. To compute the higher-order terms in (2) we need to compute the coefficients, that (k) is the partial derivatives ∂t Q(x, t) at x = 0, t = 0+ . If these were available on both sides of the initial discontinuity at x = 0, then one could implement a fairly direct approach to the evaluation of the higher order terms. The method presented below relies on the availability of all spatial derivatives rather than temporal derivatives away from the interface, see Fig. 1. In order to express all time derivatives as functions of space derivatives we apply the Cauchy-Kowalewski method and use the fact that both the physical flux and source term are the functions of the vector of conservative variables. This yields the following expressions for time derivatives: (k)

∂t Q(x, t) = P(k) (∂x(0) Q, ∂x(1) Q, . . . , ∂x(k) Q) .

(5)

These time-partial derivatives at x = 0 for t > 0 have a meaning if the spatial derivatives ∂x(0) Q, ∂x(1) Q, . . . , ∂x(k) Q can be given a meaning at x = 0 for t > 0. For x < 0 and for x > 0 all spatial derivatives ∂x(k) QL (x) , ∂x(k) QR (x) , k = 1, 2, . . . , K are defined and readily computed. At x = 0, however, we have the one-sided derivatives ∂x(k) QL (0) = lim ∂x(k) QL (x), x→0−

∂x(k) QR (0) = lim ∂x(k) QR (x), x→0+

3

k = 1, 2, . . . , K .

V.A. Titarev and E.F. Toro

We thus have a set of K pairs (∂x(k) QL (0), ∂x(k) QR (0)) of constant vectors that could be used as the initial condition for K conventional Riemann problems, if in addition we had a set of corresponding evolution equations for the quantities ∂x(k) Q(x, t). The sought evolution equations can be easily constructed. It can be verified that the quantity ∂x(k) Q(x, t) obeys the following system of non-linear inhomogeneous evolution equations ∂t (∂x(k) Q(x, t)) + A(Q)∂x (∂x(k) Q(x, t)) = Hk .

(6)

where the coefficient matrix A(Q) is precisely the Jacobian matrix of system (1). Equations (6) are obtained by manipulating derivatives of (1). The source term Hk on the right hand side of (6) Hk = Hk (∂x(0) Q(x, t), ∂x(1) Q(x, t), . . . , ∂x(k) Q(x, t)) is a function of the spatial derivatives ∂x(k) Q(x, t), for k = 0, 1, . . . , k, and vanishes when the Jacobian matrix A is constant and S ≡ 0, that is, when the original system in (1) is linear and homogeneous with constant coefficients. In order to easily solve these evolution equations we perform two simplifications, namely, we first neglect the source terms Hk and then we linearize the resulting homogeneous equations. Neglecting the effect of the source terms Hk is justified, as we only need ∂x(k) Q(x, t) at the first-instant interaction of left and right states. We thus have homogeneous nonlinear systems for spatial derivatives. Then we perform a linearization of the homogeneous systems about the leading term of the power series expansion (2), that is the coefficient matrix is taken as the constant matrix (0)

ALR = A(Q(0, 0+ )) . Thus, in order to find the spatial derivatives at x = 0, t = 0+ we solve the following homogeneous, linearized conventional Riemann problems (0)

∂t (∂x(k) Q(x, t)) + ALR ∂x (∂x(k) Q(x, t)) = 0 ,  (k)  ∂x QL (0) ,

∂x(k) Q(x, 0) = 

x 0 . (0)

            

(7)

Note that the (constant) Jacobian matrix ALR is the same coefficient matrix for all ∂x(k) Q(x, t)) and is evaluated only once, using the leading term of the expansion. We denote the similarity solution of (7) by D(k) (x/t). In the computation of all higher order terms, the solutions of the associated Riemann problems are analytic and the question of choosing a Riemann solver does not arise. The relevant value at the interface is obtained by evaluating this vector at x/t = 0, namely ∂x(k) Q(0, 0+ )) = D(k) (0) . 4

V.A. Titarev and E.F. Toro

We call this value the Godunov state, in analogy to the interface state (4) associated with the leading term. Having evolved all space derivatives at the interface x = 0 we form the time derivatives and finally define the solution of the DRPK as the power series expansion QLR (τ ) = C0 + C1 τ + C2 τ 2 + . . . + CK τ K .

(8)

where the coefficients are given by (k)

Ck = 3

∂t Q(0, 0+ )) . k!

(9)

Riemann Solvers for the Leading Term of DRP

As has already been mentioned, for complex nonlinear systems it may be difficult to find the leading term of the Taylor series expansion (2), the Godunov state, since it requires the use of a non-linear Riemann solver, exact or approximate. Here, we suggest that the recently-proposed EVILIN Riemann solver14 be used to obtain the Godunov state of the nonlinear Riemann problem (2). The computation of the Godunov state by the EVILIN Riemann solver consists of two main steps. The first step is to open the Riemann fan by using the generalized Multi-Stage (GMUSTA) Riemann solver.12, 17 The GMUSTA Riemann solver solves the local Riemann problem (3) numerically rather than analytically by means of a simple first-order scheme applying transmissive boundary conditions at each local time step. This is equivalent to evolving in time the initial data QL (0), QR (0) via the governing equations. In the second step one applies a linearized Riemann solver on the evolved initial data obtained from the GMUSTA procedure giving a close-form expression for the Godunov state. Below we briefly outline the GMUSTA and EVILIN Riemann solvers. Assume that at the initial time t = 0 we know the left and right initial data values QL (0), QR (0) of the Riemann problem (3). We introduce a separate spatial domain and the corresponding mesh with 2M cells: −M + 1 ≤ m ≤ M and cell size δx. The boundary between cells m = 0 and m = 1 corresponds to the interface position x = 0 in (3). Transmissive boundary conditions are applied at the numerical boundaries x±M +1/2 on the grounds that the Riemann - like data extends to ±∞. We now want to solve this Riemann problem numerically on a given mesh and construct a sequence of evolved data states (k) (k) Q(l) m , 0 ≤ l ≤ k in such a way, that the final values adjacent to the origin Q0 , Q1 are close to the sought Godunov state. Here k is the total number of GMUSTA time steps, or stages of the algorithm. In short, the GMUSTA time marching for m = −M + 1, . . . M is organized as follows: (l+1) = Q(l) Qm m −

´ δt ³ (l) (l) Fm+1/2 − Fm−1/2 , δx

(l)

(l)

Fm+1/2 = FGF (Q(l) m , Qm+1 ).

(10)

Here FGF is the monotone first order GFORCE numerical flux17 which is the upwind 5

V.A. Titarev and E.F. Toro

generalization of the centred FORCE flux13 and is given by: FGF = ΩFLW + (1 − Ω)FLF ,

Ω=

1 , 1 + Cg

(11)

where FLW and FLF are the centred Lax-Wendroff and Lax-Friedrichs fluxes, respectively. The GMUSTA Courant number coefficient 0 < Cg < 1 is prescribed by the user. Since the linear stability limit of the GFORCE scheme is unity, we typically take Cg = 0.9. The GMUSTA time step δt is computed as δt = δx/Smax and is then used in the time update and for evaluation FLW and FLF . Here Smax is the speed of the fastest wave in the local solution. The cell size δx can be chosen arbitrary due to the self-similar structure of the solution of the conventional Riemann problem. We usual take δx ≡ 1. We remark that although expression (11) involves centred fluxes, the resulting GFORCE flux is upwind due to the fact that the nonlinear weight Ω in (11) depends on the local wave speed. In the special case of the linear constant coefficient equation the GFORCE flux is identical to the Godunov upwind flux. The time marching procedure is stopped when the desired number of stages k is reached. At the final stage we have a pair of values adjacent to the interface position. For the construction of Godunov-type advection schemes one needs a numerical flux at the origin, which for the outlined procedure is given by (k)

(k)

GF FGM (Q(k) i+1/2 = Fm+1/2 = F m , Qm+1 ).

(12)

For the purpose of solving the derivative Riemann problem, however, we need the Godunov (k) (k) state as well. In general, the states adjacent to the origin, namely Q0 ,Q1 are different. We now use a linearized Riemann solver to resolve the discontinuity in Q at the origin resulting in the EVILIN Riemann solver.14 To this end we solve exactly the following linearized Riemann problem: (k)

  Q(k)) 0 Q(x, 0) = (k)) 

Q1

(k)

A1/2 = A( 12 (Q0 + Q1 ))

∂t Q + A1/2 ∂x Q = 0, if

x0.

(13)

We remark that conventional linearized Riemann solvers have two major deficiencies. Firstly, they give a large unphysical jump in all flow variables near sonic points, a rarefaction shock, unless explicit entropy fixes are enforced. This is due to the fact that linearized Riemann solvers do not open the Riemann fan when the solution contains a sonic point and produce instead a rarefaction shock. Secondly, they cannot handle the situation when the Riemann problem solution contains very strong rarefaction waves. These problems do not occur for the EVILIN Riemann solver, which is essentially due to the fact that we apply the linearized Riemann solver to evolved values rather than to the initial data. See14 for more details and numerical examples. 6

V.A. Titarev and E.F. Toro

It can be shown numerically17 that when the number of cells 2M and number of stages k are large, the GMUSTA flux converges to the Godunov flux with the exact Riemann solver. Correspondingly, the approximate Godunov state produced by the EVILIN solver (13) converges to the exact Godunov state, even for nonlinear systems with a complex wave pattern. For the linear constant coefficient equations this property is exact, whereas for nonlinear systems it can be verified by numerical experiments. We note that since the solution of the piece-wise constant Riemann problem (3) is self-similar, the value of the cell size δx used in the GMUSTA time marching does not influence the resulting GMUSTA and EVILIN solutions. For a given CFL number Cg these solutions depend only on the number of stages k and domain size 2M . Moreover, when M > k the transmissive boundary conditions do not affect the numerical solution of (3) which in this case depends only on k and Cg . 4

Use of the DRP in the ADER approach

The DRP Riemann solver described in the previous sections can be used to construct very high-order numerical ADER-type fluxes to be used in ADER schemes.11 In this section we review the method for the two-dimensional structured mesh case. Consider the following two-dimensional nonlinear system of conservation laws: ∂t Q + ∂x F(Q) + ∂y G(Q) = S(x, y, t, Q).

(14)

Integration of (14) over a space-time control volume produces the following one-step finitevolume scheme: ´ ´ ∆t ³ ∆t ³ Qn+1 = Qnij + Fi−1/2,j − Fi+1/2,j + Gi,j+1/2 − Gi,j−1/2 + ∆tSij (15) ij ∆x ∆y where Qnij is the cell average of the solution at time level tn : 1 1 Z xi+1/2 Z yj+1/2 n Q(x, y, tn ) dy dx, Qij = ∆x ∆y xi−1/2 yj−1/2

(16)

Fi+1/2,j , Gi,j+1/2 are the space-time averages of the physical fluxes at the cell interfaces: Fi+1/2,j = Gi,j+1/2

1 1 Z yj+1/2 Z tn+1 F(Q(xi+1/2 , y, τ )) dτ dy, ∆t ∆y yj−1/2 tn

1 1 Z xi+1/2 Z tn+1 = G(Q(x, yi+1/2 , τ )) dτ dx, ∆t ∆x xi−1/2 tn

(17)

and SN ij is the time-space average of the source term: Snij

1 1 1 Z xi+1/2 Z yj+1/2 Z tn+1 = S(Q, x, y, τ )) dτ dy dx. ∆t ∆x ∆y xi−1/2 yj−1/2 tn

Here τ = t − tn is local time. 7

(18)

V.A. Titarev and E.F. Toro

The evaluation of the ADER numerical flux Fi+1/2,j consists of the following steps. First we discretize the spatial integrals over the cell faces in (17) using a tensor product of a suitable Gaussian numerical quadrature. The expression for the numerical flux in the x coordinate direction then reads Fi+1/2,j =

Ã

N X α=1

!

1 Z tn+1 F(Q(xi+1/2 , yα , τ ))dτ Kα , ∆t tn

(19)

where yα are the integration points over the cell side [yj−1/2 , yj+1/2 ] and Kα are the weights. Normally, we use the two-point Gaussian quadrature for third and fourth order schemes and a higher-order Gaussian quadrature for fifth and higher order schemes. Next we reconstruct the point-wise values of the solution and all derivatives up to order r − 1 from cell averages at the Gaussian integration points (xi+1/2 , yα ). To avoid spurios oscillations, a non-linear solution adaptive reconstruction must be used. In this paper we use the dimension-by-dimension WENO reconstruction. For general information on reconstruction in the context of the two-dimensional ENO and WENO schemes see.1, 9 Extension to three space dimensions in the context of ADER schemes can be found in.18 After the reconstruction is carried out for each Gaussian integration point yα at the cell face we pose the Derivative Riemann problem (1) in the x-coordinate direction (normal to the cell boundary) and obtain a high order approximation to Q(xi+1/2 , yα , τ ) for the interface state at the Gaussian integration point (xi+1/2 , yα , zβ ). The only difference which occurs in the multidimensional case is that the Cauchy-Kowalewski procedure procedure will now involve mixed x, y and z derivatives up to order r − 1. The flux of the state-expansion ADER scheme is obtained by inserting the approximate state into formula (19) and using an appropriate rth -order accurate quadrature for time integration: N X

ÃN X

α=1

l=1

Fi+1/2,jk =

!

F(Q(xi+1/2 , yα , τl ))Kl Kα .

(20)

For the flux expansion ADER schemes we write Taylor time expansion of the physical flux " # at each point (xi+1/2 , yα , zβ ) r−1 X ∂k τk F(xi+1/2 , yα , τ ) = F(xi+1/2 , yα , 0+) + F(x , y , 0+) . (21) i+1/2 α k k! k=1 ∂t The numerical flux is given by Fi+1/2,jk =

N X α=1

Ã

F(xi+1/2 , yα , 0+) +

r−1 X k=1

"

#

!

∂k ∆tk F(xi+1/2 , yα , 0+) Kα . ∂tk (k + 1)!

(22)

The leading term F(xi+1/2 , yα , 0+) is computed from (1) using a monotone upwind flux. The remaining higher order time derivatives of the flux in (21) are expressed via time derivatives of the intercell state Q(xi+1/2 , yα , τ ) which are given by the Taylor expansion for the interface state. 8

V.A. Titarev and E.F. Toro

The computation of the numerical source is straightforward and involves three-dimensional integrals. First we use the tensor-product of the N -point Gaussian rule to discretize the two-dimensional space integral in (18) so that the expression for Sij reads Sij =

à N X N X 1 Z tn+1 α=1 β=1

∆t

tn

!

S(xα , yβ , τ, Q(xα , yβ , zγ , τ ))dτ

Kβ Kα .

(23)

Then we reconstruct values and all spatial derivatives, including mixed derivatives, of Q at the Gaussian integration point in x − y − z space for the time level tn . Note that these points are different from flux integration points over cell faces. The reconstruction procedure is entirely analogous to that for the flux evaluation. Next for each Gaussian point (xα , yβ ) we perform the Cauchy-Kowalewski procedure and replace time derivatives by space derivatives. As a result we have high-order approximations to Q(xα , yβ , τ ). Finally, we carry out numerical integration in time using the Gaussian quadrature: Sij =

N X N X

ÃN X

α=1 β=1

l=1

!

S(xα , yβ , τl , Q(xα , yβ , τl ))Kl Kβ Kα .

(24)

The solution is advanced in time by updating the cell averages according to the one-step formula (15). 5

Applications

In this section we show some results of the ADER schemes with the conventional stateexpansion formulation, denoted as ADER-AD, and with the new EVILIN-based variant of the DRP solver, denoted by ADER-GM. We first illustrate the performance of the conventional ADER schemes for the hyperbolic systems with source terms. 6

Shallow water equations

We consider the case of horizontal channels with rectangular cross section. The twodimensional shallow-water equations with pollutant transport have the form (14) with S = S1 + S2 and    

Q=

h hu hv hψ





    , F =   

hu 2 hu + 21 gh2 huv huψ





    , G =   

hv hvu hv 2 + 21 gh2 hvψ

    , 

   

S1 = 

0 −gbx h −gby h 0

    (25) 

Here u and v are respectively x and y components of velocity, h is the depth, b is the bed elevation, g = 9.8 is acceleration due to the gravity and ψ is concentration of pollutant. We consider a time-dependent flow with periodic boundary conditions in a domain [−5 : 5]2 . We take the exact solution in the following form:

9

V.A. Titarev and E.F. Toro

    

h hu hv hφ





    =  

5 + sin πt sin πx sin πy (5 + sin πt sin πx sin πy) sin πt sin πx (5 + sin πt sin πx sin πy) sin πt sin πy (5 + sin πt sin πx sin πy) sin πt sin πx sin πy

    

(26)

The bottom elevation has the form: b(x, y) = exp (−α(x2 + y 2 ), with α = 10. The second part of the source term S2 is adjusted in such a way that we have the exact solution given above. Method ADER3

ADER4

Mesh L∞ error 25 × 25 2.82 × 10−2 50 × 50 8.53 × 10−3 100 × 100 4.54 × 10−4

L∞ order

25 × 25 6.92 × 10−3 50 × 50 2.06 × 10−3 100 × 100 7.15 × 10−5

1.73 4.23

L1 error 7.55 × 10−1 2.33 × 10−1 4.26 × 10−3

1.75 4.85

3.35 × 10−1 7.31 × 10−2 1.49 × 10−3

L1 order 1.69 5.78

2.19 5.61

Table 1: Convergence test for ADER schemes as applied to equations (25) with the exact solution (26).

Table 1 shows the convergence study for the output time t = 1 for ADER3 and ADER4 schemes The errors of cell averages of the solution in L∞ and L1 norms are presented. We observe that approximately fifth order of accuracy is achieved by all schemes. It is interesting to note that these orders of accuracy actually exceed the fourth order accuracy of the two-point Gaussian rule used for flux integration. 6.1

Compressible Euler Equations

The two-dimensional compressible Euler equations correspond to the following choice in (14)    

Q=

ρ ρu ρv E

   , 

   

F = Qu + 

0 p 0 pu

   , 

   

G = Qv + 

0 0 p pv

    

(27)

1 p = (γ − 1)(E − ρ(u2 + v 2 )) 2 and S = 0. Here ρ, u, v, p and E are density, components of velocity in the x and y coordinate directions, pressure and total energy, respectively; γ is the ratio of specific heats. We use γ = 1.4 throughout.

10

V.A. Titarev and E.F. Toro

Method

Mesh

L∞ error

L∞ order

L1 error

L1 order

ADER3-AD11

25 × 25 50 × 50 100 × 100

1.97 × 10−1 2.20 × 10−2 7.39 × 10−4

3.16 4.90

1.166 8.58 × 10−2 3.39 × 10−3

3.77 4.66

25 × 25 50 × 50 100 × 100

2.04 × 10−1 2.51 × 10−2 8.37 × 10−4

3.02 4.90

1.205 9.20 × 10−2 3.65 × 10−3

3.71 4.66

25 × 25 50 × 50 100 × 100

3.14 × 10−1 4.04 × 10−2 1.36 × 10−3

2.96 4.89

1.653 1.88 × 10−1 1.07 × 10−2

3.13 4.14

ADER3-GM

WENO9

Table 2: Density convergence study for the vortex evolution problem (28) at the output time t = 50. Schemes with piece-wise parabolic reconstruction.

6.1.1

Two-dimensional vortex evolution problem

We solve the two-dimensional Euler equations in the square domain [−5 : 5] × [−5 : 5] with periodic boundary conditions. The initial condition corresponds to a smooth vortex placed at the origin and is defined as the following isentropic perturbation to the uniform flow of unit values of primitive variables:9 ε 1 (1−r2 ) ε 1 (1−r2 ) u=1− e2 y, v = 1 + e2 x, 2π 2π (28) p (γ − 1)ε2 (1−r2 ) e , = 1, T =1− 8γπ 2 ργ where r2 = x2 + y 2 and the vortex strength is ε = 5. The exact solution is a vortex movement with a constant velocity at 45o to the Cartesian mesh lines. We compute the numerical solution at the output time t = 50 (five periods) using Ccf l = 0.45 for all runs. We limit ourselves to the schemes with piece-wise parabolic (r = 3) polynomials. We also run the state-of-the art finite-volume WENO schemes9 for comparison purposes. The ADER schemes use the DRP solver to obtain the numerical flux and two-point Gaussian quadrature for spatial integration in the numerical flux, whereas the WENO scheme uses the Rusanov-type numerical flux7 as the building block, three-point Gaussian quadrature and third-order TVD RK method10 for the temporal update. The resulting ADER methods (ADER-AD, ADER-GM, WENO) are of fourth order spatial accuracy whereas the WENO scheme is fifth-order accurate in space due to the sixth order Gaussian quadrature. All methods are third-order accurate in time. Since the reconstruction step is essentially the same for all methods, the difference in accuracy can result only from the temporal discretization and the numerical fluxes. 11

V.A. Titarev and E.F. Toro

1

0.8

Y

0.6

0.4

0.2

0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

1.75

2

2.25

2.5

2.75

3

X 1

0.8

Y

0.6

0.4

0.2

0 0

0.25

0.5

0.75

1

1.25

1.5

X

Figure 1: Density convergence study for the double Mach reflection problem.

0.4

0.4

0.3

0.3

Y

0.5

Y

0.5

0.2

0.2

0.1

0.1

0

0 2

2.25

X

2.5

2.75

2

2.25

X

2.5

2.75

Figure 2: Density convergence study for the double Mach reflection problem. Zoom of the blown-up region of Fig. 1.

Table 2 shows the convergence study for all schemes. We present errors and convergence rates in L∞ and L1 norms for cell averages of density. Overall, we see that all methods achieve the designed order of accuracy. Both ADER schemes compare well with the

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finite-volume WENO scheme. 6.1.2

Double Mach reflection of a strong shock

The double Mach reflection problem19 is a standard test problem for testing robustness and accuracy of advection schemes. The setup of the problem, initial conditions and description of the flow physics can be found in.19 The problem has been studied intensively in recent years, see e.g.2, 9 and references therein. The results of the conventional fluxexpansion ADER schemes can be found in11 and are not shown here. Figs. 1,2 show numerical results of the new ADER3-GM scheme for two meshes: 960×240 cells (top) and 1920 × 480 cells (bottom). We see that the scheme produces the flow pattern generally accepted in the existing literature as correct. All discontinuities are well resolved and correctly positioned. The overall accuracy of our new ADER3-GM scheme compares well with that of the WENO scheme9 and conventional state-expansion ADER schemes.11 6.1.3

Shock-vortex interaction problem

For the detailed description of the problem see6 and references therein. The initial data consists of a stationary shock wave of Mach number Ms = 1.5, positioned at x = 0.5 inside the computational domain [0 : 2] × [0 : 1] and a moving composite vortex of the Mach number Mv = 0.9. As the vortex moves to the right of the domain, a complex interaction pattern emerges containing a few shocks as well as a rich smooth structure. Therefore, this test problem is ideal for very high-order non-oscillatory methods. Figs. 3, 4 show the density plots for the second output time t = 0.69 from6 and the third order ADER-AD and ADER3-GM schemes on a mesh of 800×401 cells. We see that our ADER3-AD solution agrees well with that reported in the literature with all features very well resolved. In fact, the ADER3-AD solution appears to be more accurate than that of.6 On the other hand, the use of the GMUSTA flux in the ADER3-GM scheme produces a start-up error which manifests itself as a spurious acoustic wave moving to the right. This is probably due to the fact that the GMUSTA Riemann solver does not resolve the stationary shock profile exactly. Indeed, the shock-wave front is much sharper in the ADER3-AD solution. We remark that start-up errors can be cured in a standard way. 6.2

Computational cost and memory requirements

Table 3 shows approximate computing times for the schemes with piece-wise parabolic reconstruction as applied to the double Mach reflection problem. All numbers are normalized by the computing time of the ADER3-AD scheme. We see that the third-order ADER schemes are faster than the WENO scheme with the Rusanov flux roughly by 66% for the state expansion version and for 40% for the flux expansion version with the EVILIN flux. The reason for this is twofold. Firstly, our scheme needs to perform the 13

V.A. Titarev and E.F. Toro

Figure 3: Density convergence study for the shock-vortex interaction problem. ADER3-AD scheme on a mesh of 800 × 401 cells.

Figure 4: Density convergence study for the shock-vortex interaction problem. ADER3-GM scheme on a mesh of 800 × 401 cells.

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very costly characteristic projections and smoothness indicators computations in the reconstruction procedure only once during one time step. Secondly, the ADER scheme uses the two-point integration rule to evaluate the numerical fluxes whereas the WENO scheme in the cited reference uses the three-point Gaussian rule. Method ADER3-AD ADER3-GM WENO,9 Rusanov flux

Normalized computing time 100% 112% 166%

Table 3: Approximate computation times for the double Mach reflection problem.

Next, we discuss the memory requirement of the schemes. The ADER schemes of any order effectively need only two global arrays to store the vector of the conservative variables and the total sum of fluxes. The WENO schemes with the third order three-stage TVD Runge-Kutta method need at least three such arrays. Note that expensive memory transfers may be needed for the RK method in this case. Higher-order RK methods, e.g. fourth order, will have substantially higher memory requirements. 7

Conclusions

In this paper we have presented a modification of the solution procedure for the derivative Riemann problem which does not require an approximate-state Riemann solver. Our new DRP solver extends very high order upwind schemes to a large class of hyperbolic systems of conservation laws for which the Riemann problem solution is not available. We implemented the new Derivative Riemann solver in the framework of finite-volume ADER schemes in multiple space. The presented numerical results illustrate the very high order of accuracy as well as the essentially non-oscillatory property of the ADER schemes with conventional and DRP solvers. When compared with the state-of-art finite-volume WENO schemes, the ADER schemes are faster, more accurate and need a smaller amount of computer memory. Acknowledgements. The first author acknowledges the financial support provided by the PRIN programme (2004-2006) of Italian Ministry of Education and Research (MIUR). The second author acknowledges the support of an EPSRC grant, as senior visiting fellow (Grant GR N09276) at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2003. REFERENCES [1] J. Casper and H. Atkins. A finite-volume high order ENO scheme for two dimensional hyperbolic systems. J. Comput. Phys., 106:62–76, 1993. 15

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[2] B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16:173–261, 2001. [3] M. Dumbser and C.-D. Munz. Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comp., 2005. to appear. [4] S.K. Godunov. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sbornik, 47:357–393, 1959. [5] G.S. Jiang and C.W. Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126:202–212, 1996. [6] A. Rault, G. Chiavassa, and R. Donat. Shock-vortex interactions at high Mach numbers. J. Sci. Comp., 19(1-3):347–371, 2003. [7] V.V. Rusanov. Calculation of interaction of non-steady shock waves with obstacles. USSR J. Comp. Math. Phys., 1:267–279, 1961. [8] T. Schwartzkopff, M. Dumbser, and C.D. Munz. Fast high order ADER schemes for linear hyperbolic equations. J. Comput. Phys., 197(2):532–539, 2004. [9] J. Shi, C. Hu, and C.-W. Shu. A technique for treating negative weights in WENO schemes. J. Comput. Phys., 175:108–127, 2002. [10] C.-W. Shu. Total - variation - diminishing time discretizations. SIAM J. Scientific and Statistical Computing, 9:1073–1084, 1988. [11] V.A. Titarev and E.F. Toro. ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys., 204(2):715–736, 2005. [12] V.A. Titarev and E.F. Toro. MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. International Journal for Numerical Methods in Fluids, 49(2):117–147, 2005. [13] E.F. Toro. Riemann solvers and numerical methods for fluid dynamics. SpringerVerlag, 1999. Second Edition. [14] E.F. Toro. Riemann solvers with evolved initial conditions. Preprint NI05003-NPA. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2005. Also Int. J. Numer. Meth. Fluids, to appear. [15] E.F. Toro, R.C. Millington, and L.A.M. Nejad. Towards very high order Godunov schemes. In E. F. Toro, editor, Godunov Methods. Theory and Applications, pages 907–940. Kluwer/Plenum Academic Publishers, 2001.

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[16] E.F. Toro and V.A. Titarev. Solution of the generalised Riemann problem for advection-reaction equations. Proc. Roy. Soc. London, 458 (2018):271–281, 2002. [17] E.F. Toro and V.A. Titarev. MUSTA schemes for systems of conservation laws. Preprint NI04033-NPA. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2004. Also J. Comp. Phys., to appear. [18] E.F. Toro and V.A. Titarev. ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions. J. Comput. Phys., 202(1):196– 215, 2005. [19] P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54:115–173, 1984.

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