Department of Computing and Mathematics, Manchester Metropolitan University, Chester Street, Manchester, M1 5GD, U.K. s:

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1 When citing this article please use: E F Toro, R C Millington and L A M Nejad. Towards very high-order Godunov schemes. In Godunov Methods: Theory and Applications. Edited Review. E F Toro (Editor). Kluwer Academic/Plenum Publishers. Conference in Honour of S K Godunov. Vol., pages New York, Boston and London,. TOWARDS VERY HIGH ORDER GODUNOV SCHEMES E. F. TORO, R. C. MILLINGTON AND L. A. M. NEJAD Department of Computing and Mathematics, Manchester Metropolitan University, Chester Street, Manchester, M 5GD, U.K. s: E.F.Toro@doc.mmu.ac.uk SUMMARY. We present an approach, called ADER, for constructing non-oscillatory advection schemes of very high order of accuracy in space and time; the schemes are explicit, one step and have optimal stability condition for one and multiple space dimensions. The approach relies on essentially non-oscillatory reconstructions of the data and the solution of a generalised Riemann problem via solutions of derivative Riemann problems. The schemes may thus be viewed as Godunov methods of very high order of accuracy. We present the ADER formulation for the linear advection equation with constant coefficients, in one and multiple space dimensions. Some preliminary ideas for extending the approach to non-linear problems are also discussed. Numerical results for one and two-dimensional problems using schemes of upto -th order accuracy are presented.. Introduction We are concerned with the construction of non-oscillatory numerical schemes of very high-order of accuracy for hyperbolic conservation laws. Here we present a new approach, called ADER, for constructing such schemes. The ADER formulation is presented for the linear advection equation with constant coefficients in one, two and three space dimensions; schemes for linear hyperbolic systems with constant coefficients in one and multiple space dimensions can then be easily constructed. As is well-known from Godunov s

2 E. F. TORO ET AL. theorem(godunov, 959) high accuracy and absence of spurious oscillations near discontinuities are contradictory requirements on numerical methods; the theorem says that any (linear) scheme of accuracy greater than one will be oscillatory near discontinuities. A classical way of circumventing Godunov s theorem is to construct non-linear schemes, even when applied to linear problems. A successful class of non-linear schemes are the so-called total Variation Diminishing Methods, or TVD methods, (Harten, 983), (Sweby, 984) developed over the last two decades or so. Such schemes provide today the basis of a mature numerical technology suitable for industrial and scientific applications; for upto date presentations of these methods see for example (LeVeque, 99), (Toro, 997), (Toro, 999), (Kröner, 997), (Godlewski and Raviart, 996) and (Hirsch, 988). These methods are second-order accurate almost everywhere, reduce locally to first-order of accuracy and are known to be unsuitable for special application areas such compressible turbulence and for wave propagation problems involving long-time evolution; extrema are clipped and numerical dissipation may become dominant. The aim of this paper is to present an approach capable of producing schemes of uniform very high accuracy for smooth solutions and non-oscillatory near discontinuities, without imposing TVD-like constraints. There are at present several approaches for constructing high-order methods that are applicable to hyperbolic conservations laws. Examples include spectral methods (Canuto and Quarteroni, 99), discontinuous Galerkin finite element methods (Cockburn and Shu, 998), (van der Vegt, ), the class of compact difference schemes of Tolstykh and co-workers (Tolstykh, 994) and the UNO and ENO/WENO schemes of Harten and co-workers (Harten and Osher, 987), (Harten et al., 987), (Shu, 997). The requirement of high-order of accuracy is met by all of the above approaches, although there may be problems in preserving time accuracy, particularly for multiple space dimensions and problems involving source terms. The ENO schemes enjoy an extra advantage; they fulfil the requirement of absence of spurious oscillations near discontinuities in a way that is described as essentially non-oscillatory. This means that solutions to model problems are, to the eye, free from spurious oscillations and that such oscillations can be shown to decay as the mesh is refined. We note that this

3 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 3 is not a property enjoyed by conventional, linear schemes. In this paper we present a new approach for constructing very highorder non-oscillatory schemes for hyperbolic conservation laws. The approach is related to the ENO/WENO methods and is so far applicable to linear hyperbolic systems with constant coefficients in one and multiple space dimensions. The schemes are conservative, one-step, explicit and fully discrete, requiring only the computation of the inter-cell fluxes to advance the solution by a full time step. For a one-dimensional problem for example, to compute the inter-cell numerical flux we solve a sequence of m Riemann problems for the k-th spatial derivatives of the solution, with k =,,...,m, where m is arbitrary and is the order of accuracy of the resulting scheme. The case m = reproduces the basic Godunov first-order upwind method. The non-linear (non-oscillatory) version of the schemes results from applying ENO polynomial reconstructions, in much the same way as in the ENO schemes. The rest of this paper is organised as follows. In section we review the GRP and Modified GRP schemes, whose philosophy serves as the inspiration for the ADER approach; in section 3 we present the basis of the ADER approach for the one-dimensional case. In section 4 we present ADER for two and three-dimensional linear problems; in section 5 we discuss some preliminary ideas for extending ADER to non-linear problems. In section 6 we present some numerical experiments for one and two-dimensional examples. Section 7 summarises the paper and discusses possible future developments.. Review of the GRP Scheme The Generalised Riemann problem (GRP) scheme of Ben-Artzi and Falcovitz (Ben-Artzi and Falcovitz, 984) and a modification to it proposed by Toro in 99 (unpublished) serve as the inspiration for the construction of the ADER schemes presented in this paper. We therefore review this approach and the appropriate modifications that are necessary for the construction of very high order schemes.

4 4 E. F. TORO ET AL... BACKGROUND Consider the scalar conservation law t q + x f(q) =, () whereq(x,t) is the conserved variable and f(q) is thephysical fluxfunction. The integral form of (), which admits discontinuous solutions, is (qdx f(q)dt) =, () which under suitable smoothness assumptions reproduces the differential form () of the conservation law. Consider now a control volume in x-t space of dimensions x t. Evaluation of the integral form () in this control volume produces the conservative formula q n+ i = qi n t [ f x i+ f i ], (3) where q n i is the integral average qi n = xi+ q(x,t n )dx (4) x x i at time t = t n in the volume [ I i = x i,x i+ ] (5) of width x = x i+ x i. (6) The fluxes in (3) are interpreted as time averages of the physical flux, namely f i+ = t t f(q(x i+,t)) dt. (7) Conservative numerical methods for () are based on (3), whereby an expression for the numerical flux f i+ is provided. Godunov (Godunov, 959) introduced the idea of computing the numerical flux in (3) by evaluating (7) in terms of the solution q (x/t) of a local initial value problem (IVP)

5 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 5 called the Riemann problem t q + x f(q) = qi n if x < q(x,) = qi+ n if x > (8) evaluated at the cell interface x i+. Conventionally, the initial data q(x, ) in the Riemann problem is piece-wise constant and consists of two constant states of the form (4) separated by a discontinuity. Such distribution of the initial data is traditionally associated with a first-order accurate Godunovtype scheme with numerical flux f i+ = f(q ()) (9) obtainedbyevaluating(7)withintegrandf(q (x/t)).second-ordergodunovtype schemes can be constructed in essentially two ways. One approach is provided by the the Weighted Average Flux Method of Toro (Toro, 989), (Billett and Toro, 997) whereby the same piece-wise constant data Riemann problem of the first-order Godunov method is solved but this time its solution is utilised more fully by taking a space/time integral of it in an appropriate control volume. Another approach for constructing second-order Godunov methods relies on piece-wise linear reconstruction of the data and the solution of a so called generalised Riemann problem at each inter-cell position; examples of schemes in this class are the piece-wise linear method (PLM) of Colella (Colella, 985) and the MUSCL-Hancock scheme (van Leer, 985). In this case the Riemann problem reads t q + x f(q) = q i (x) if x < () q(x,) = q i+ (x) if x > where q i (x) and q i+ (x) are linear functions in x. IVP () is called the Generalised Riemann Problem, or GRP. In both the Colella and MUSCL- Hancock schemes one resorts to simplifications of the solution of this GRP, in which the so called boundary extrapolated values q L = lim q i (x), q R = lim q i+ (x) () x x i+ x x + i+

6 6 E. F. TORO ET AL. play a crucial role. In the PLM scheme for instance, these values constitute the initial condition for a piece-wise constant data Riemann problem, such as (8) with initial data (), to provide one term of the flux, with the rest of the information computed from tracing characteristics back to the reconstructed data. We note here that using only the first term for the flux leads to an unstable scheme. In the MUSCL-Hancock scheme the boundary extrapolated values () are evolved by half the time step before solving a conventional piece-wise constant data Riemann problem such as (8) with initial data given by evolved boundary extrapolated values. See Toro (Toro, 998), (Toro, 999) for details... CONVENTIONAL AND MODIFIED GRP Another approach for constructing second-order Godunov-type methods is the GRP approach of Ben-Artzi and Falcovitz (Ben-Artzi and Falcovitz, 984), in which the generalised Riemann problem () is solved more directly than in the PLM and MUSCL-Hancock schemes. Assuming the reconstruction in () is linear, Ben-Artzi and Falcovitz produce an expression for the solution of the generalised Riemann problem at the interface x = x i+ and time t = t; they do so by use of a time Taylor series around t =, namely expansion at x = x i+ q n+ i+ = q () + i+ t ( tq) i+ +O( t ), () where the term q () i+ at time t = accounts for the very first effect of the interaction of the two piece-wise linear states in (). Such first interaction is solely determined by the boundary extrapolated values q L and q R in (). Thus the first term q () in the Taylor series expansion () is ob- i+ tained by solving the conventional Riemann problem (8) with initial data q L and q R given by (); this part is common to the PLM and MUSCL- Hancock schemes. Computing the second term in () in the conventional GRP method can be labourious. Ben-Artzi and Falcovitz provided an expression for this term for the non-linear system of compressible Euler equations, thus obtaining a very robust second order Godunov scheme.

7 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 7 A simplification to the GRP scheme of Ben-Artzi and Falcovitz that is central to the development of ADER was suggested by Toro in the early 9 s, whereby the computation of the second, difficult, term is replaced by that of solving a linear Riemann problem for the gradient of the solution, as follows. First we re-write the conservation law () in quasi-linear form t q +λ(q) x q =, (3) where λ(q) = df (4) dq is the characteristic speed; for systems this is the Jacobian matrix. Then, by use of (3) we replace the time derivative in () by a space derivative so that the expansion (), to second order, reads q n+ i+ = q () i+ t (λ(q)) i+ The next problem is the computation of (λ(q)) i+ ( x q) i+ is the following. First linearise (3) around the state q () i+. (5) ( x q) i+ ; one possibility in (), which is the leading term in the expansion and results from solving a conventional non-linear Riemann problem. The linearised problem reads t q + ˆλ x q =, (6) where ˆλ = λ(q () ). Then, it is easily seen that v i+ x q obeys the linearised evolution equation (6) identically. Moreover, the following more general result holds: THEOREM : Letv (k) x q bethek-thorderspatial derivativeof q(x,t). Then v obeys the linearised evolution equation (6) exactly. The proof is trivial and is thus omitted. We note that the same result is also valid for all time derivatives v (k) t q. By virtue of the above theorem we can pose the following Riemann problem for the gradient of the solution, namely t v + ˆλ x v = q () L () x q i (x) if x < (7) v(x,) = q () R () x q i+ (x) if x >

8 8 E. F. TORO ET AL. the solution of which is q () (x/t) = i+ q () L if x ˆλt < q () R if x ˆλt > (8) Finally, the inter-cell flux for the Modified GRP scheme (MGRP) is given by ( ) f (mgrp) = f q (mgrp), q (mgrp) = q () i+ i+ i+ i+ t ˆλ q (), (9) i+ where q () is the solution of the conventional Riemann problem (8) with i+ initial data() andq () is the solution (8) of the first-derivative Riemann i+ problem (7), evaluated at x/t =, the inter-cell position. The above Modified GRP scheme was extended by MSc student Cheney (Cheney, 994) to solve advection-diffusion problems such as t q + x f(q) = α () x q, () where the viscous flux α () x q was evaluated using the solution (8) of the gradient Riemann problem (7). A benefit of so doing is an enlarged stability region, in comparison to simple central differencing for the viscous term. The Modified GRP scheme was also extended by Boden(993, unpublished) to solve the non-linear two-dimensional shallow water equations. Full details of Modified GRP second-order schemes for non-linear systems are presented in (Toro, 998). MSc student Cáceres (Cáceres, 993) attempted an extension of the modified GRP scheme to third order of accuracy. It was thought at the time that it was a simple matter of using quadratic reconstructions of the data in the generalised Riemann problem and retention of three terms in the Taylor series expansion (). The derived scheme by so doing is only O( x 3, t ) and the linearised stability condition is not the optimal condition c but c 3 ( + 3).87, () where c is the Courant number c = λ t/ x. Given the loss of time accuracy and the reduction in the stability range for the attempted third order scheme, it became clear that a different expression for the inter-cell

9 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 9 flux expansion was required, if successful very high order schemes could be constructed, still using the MGRP idea. 3. ADER for One-Dimensional Problems The ADER approach is a successful attempt to exploit the Modified GRP scheme to construct schemes of very high order of accuracy and has the following main ingredients. We first produce high-order reconstructions of the initial data and pose a generalised Riemann problem in which the initial data is piece-wise smooth, possibly with a discontinuity at the inter-cell edge. The solution of this generalised Riemann problem is then Taylor expanded in time, at the interface position, to any order of accuracy. We then obtain an average ADER state by time-integrating the Taylor series expansion. A crucial step then follows, which consists of replacing time derivatives by space derivatives, as done for the Modified GRP scheme presented in the previous section; for the linear case this is a trivial step. The main task left is the evaluation of all the spatial derivatives in the expansion; this is achieved by an evolution step based on Theorem. Evolution equations for all spatial derivatives are first identified; for the linear case these happen to be homogeneous linear advection equations. To complete the evolution process we pose and solve Riemann problems for derivatives, and thus all terms in the expansion for the time average ADER state are determined. This state is then used to compute the ADER numerical flux. We now discuss details of the above ingredients. 3.. RECONSTRUCTION AND GENERALISED RIEMANN PROBLEM The data in the form of cell averages is reconstructed via high-order polynomials. Use of a fixed stencil for the reconstruction in the ADER approach leads to linear ADER schemes. This means that the schemes have constant coefficients when applied to a linear equation or to a linear system with constant coefficients. In accordance with Godunov s theorem, these linear schemes of accuracy greater than one will produce spurious oscillations in the vicinity of discontinuities. To resolve this difficulty we use the ENO (essentially non-oscillatory) interpolation procedure, first presented by Harten

10 E. F. TORO ET AL. and collaborators (Harten and Osher, 987), (Harten et al., 987), (Shu, 997) in the context of ENO schemes. This interpolation theory has by now reached a stage of maturity and it is proving useful in various areas of application; useful background on the ENO interpolation theory in found in (Laney, 998). In the ENO procedure the stencil of the interpolation is adaptive and leads to non-linear ADER schemes; the schemes are nonlinear even when applied to linear problems. Fig. depicts higher-order polynomial reconstructions q i (x) and q i+ (x) of the data for cells I i and I i+ respectively. q k (x) q i+ (x) q (x) i i x i + / i+ x Figure. Illustration of high-order polynomial reconstructions q i(x) and q i+(x) in cells I i and I i+. Functions q i(x) and q i+(x) form initial condition for a generalised Riemann problem at x = x i+. In order to compute the ADER numerical flux at the inter-cell position x i+ for a scheme of m-th order of accuracy in space and time we solve the generalised Riemann problem q(x,) = t q + x f(q) = { qi (x) if x < q i+ (x) if x > () where the initial data are the (m )-th order polynomial functions q i (x) and q i+ (x), as depicted in Fig.. We want to find an expression for the numerical flux, or for a state, at the inter-cell position; this will involve some kind of series expansion. Next we study three possible approaches for finding such expansion.

11 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 3.. STATE SERIES EXPANSION: APPROACH I Consider the general initial value problem for the linear advection equation PDE : t q +λ x q = IC : q(x,) q (x) (3) where the characteristic speed λ is constant and the initial condition q (x) is an arbitrary function of distance x. THEOREM : For a sufficiently smooth function q (x) in IVP (3), one may express the solution at a time t = t as given by the finite volume type formula q(x, t) = q(x,)+ t [λ q(x x x) λ q(x+ ] x), (4) where q(x,t) = x+ x) q(x,t)dx (5) x x x is a cell average in the interval [x x,x+ x], q(x) = q () (x)+ k= ( λ t) k (k+)! q(k) (x) (6) and q (k) (x) is the k-th order spatial derivative of q (x), namely q (k) (x) (k) x q (x). (7) Before proving the above statement, the following remarks are in order. First we note that no discretisation is involved here, the statements are exact. Expression (4) looks like a finite volume conservative scheme, of infinite accuracy, in which q(x, t) is an integral average of the solution over the interval [x x,x + x], just as in finite volume numerical methods; the sought state is (6) and the numerical flux will have the form f λ q. Manipulation of the exact solution has given us the correct expansion (6) for the sought state. We also note that (6) coincides with a Taylor expansion at t = t only for the first two terms.

12 E. F. TORO ET AL. Proof: The analytical solution of IVP (3) at a time t is given by q(x, t) = q (x λ t). (8) Now assume that q (x) is sufficiently smooth so that all its spatial derivatives are defined, then a Taylor series expansion of (8) reads q(x, t) = q (x)+ ( λ t) k q (k) (x), (9) k! k= withq (k) (x)asgivenby(7).theinfinitesummationin(9)isnowregarded as a function of x, for a fixed time step t, and may be expressed as where ( λ t) k q (k) k! k= q(x) = q () (x)+ Use of (3) and (3) in (9) gives d (x) = λ t, (3) dx q(x) k= ( λ t) k (k +)! q(k) (x). (3) q(x, t) = q(x,) λ t dx q(x) d. (3) We now integrate (3) with respect to x in the interval [x x,x+ x] and divide the result by x to produce (4), with definitions (5)-(7), and the theorem is thus proved STATE SERIES EXPANSION: APPROACH II A clearer and entirely equivalent procedure to obtain an expansion for an inter-cell state and thus the numerical flux is as follows. We first express the solution of the generalised Riemann problem () at time t = τ as a time Taylor series expansion at the cell inter-cell position x = x i+ around the initial time t = (in local coordinates) q(x i+,τ) = q(x i+,)+ k= τ k k! (k) t q(x i+,). (33)

13 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 3 We then obtain a time-average state q ader i+ interval [, t] as q ader i+ = t t by time-integrating (33) in the q(x i+,t)) dt. (34) and use the Lax and Wendroff procedure (Lax and Wendroff, 96), (Shu, 997) to substitute all time-derivatives by space-derivatives using the differential equation; for the one-dimensional linear advection equation with constant speed the time derivatives are (k) t q = ( λ) k (k) x q. (35) By retaining a finite number m of terms in the above procedure one obtains the average state expansion q ader i+ = q () m ( λ t) k + i+ (k +)! q(k) i+ k= Then, the ADER numerical flux follows as f ader i+. (36) = f(q ader i+ ), (37) which if substituted in the finite volume one-step conservative formula (3) leads to a scheme of m-th order accuracy in space and time for the linear advection equation with constant coefficient DIRECT FLUX EXPANSION Approach II above can be applied to derive the ADER flux more directly; this is possible by using a remarkable property of the flux (Ghidaglia et al, 999) of a conservation law such as (), which we now state. THEOREM 3: For any conservation law () the flux function f(q) obeys the evolution equation where λ(q) is the characteristic speed. t f(q)+λ(q) x f(q) =, (38) Proof: The proof is straightforward. Multiply () by λ(q) and use the chain rule to obtain t f(q) = λ(q) t q and the result follows.

14 4 E. F. TORO ET AL. The above property is also valid for non-linear systems, as is trivially seen by replacing the characteristic speed by the Jacobian matrix. For the linear scalar case, application of above property (38) leads to a direct expression for the ADER flux, namely f ader i+ = f () m ( λ t) k + i+ (k +)! f(k) i+ k=. (39) In this expression one could in principle use approximate Riemann solvers that give direct expressions for the flux rather than for a state. We remark here that we have performed numerical experiments in which the Godunov flux based on the Riemann problem solution has been replaced by a centred flux, such as the FORCE flux (Toro, 996), (Toro and Billett, ). The results from using upwind and centred fluxes in the ADER approach look in general comparable and for very high orders of accuracy they are indistinguishable. However these observations are only provisional EVOLUTION VIA DERIVATIVE RIEMANN PROBLEMS We now have an expression for the sought average state to be utilised in the evaluation of a numerical flux. We utilise the analytical developments of approaches I and II above in a numerical set up. Assume polynomial reconstructions q i (x) and q i+ (x) of the initial data in cells I i and I i+ respectively and formulate the generalised Riemann problem as in(). The ADER state generated by thisgrp is denoted by q ader, with corresponding i+ numerical flux f ader i+ = f(q ader i+ ) ; q ader i+ = q () m ( λ t) k + i+ (k+)! q(k) i+ k=. (4) Here r (k) i+ is the solution r (k) (x, t) of the k-th order derivative Riemann i+ problem evaluated at x/t =, as we now explain. By virtue of Theorem we know that for the linear advection equation with constant coefficients all spacial derivatives satisfy an evolution equation, namely the homogeneous

15 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 5 linear advection equation. We then define m derivative Riemann problems t v +λ x v = q (k) i if x < (4) v(x,) = q (k) i+ if x > where the initial condition for the k-th Riemann problem is given by q (k) i lim x x x (k) q(x,) i+ q (k) i+ lim x x + x (k) q(x,) i+ (4) and k =,,...,m. Note that all derivative Riemann problems are defined even if there are discontinuities at inter-cell boundaries. Thus the average ADER state (4) is completely defined and the ADER numerical flux is given as in (4). The average state in (4) has leading term q (), which corresponds to i+ the Godunov first-order upwind method. Second and higher order schemes (m ) include a function evaluated at times t k = α k t, with weight α k (k +) k, k. (43) Itis easy to see that thesequence of time weights α k lie in theinterval [,], as lim k α k =. For all times t k wethen havet k [ t, t].fig. shows a graph of the time weights α k for schemes of accuracy m =,..., RE-INTERPRETATION OF ADER SCHEMES Substitution of the numerical fluxes (4) into the conservative formula (3) produces m = qi n with q n+ i p (k) i+ k= = ( λ) k q (k) i+ ( t) k x [ ] λp (k) λp (k) i+ i, (44), ( t) k = ( t)k+ (k+)!. (45)

16 6 E. F. TORO ET AL. Time weights.5 5 Order of accuracy Figure. Variation of the time weights α k with the order of accuracy m, for m =,...,. For m = we have a single correction to the initial data qi n given by the Godunov first-order upwind method with a time step ( t) = t. For m = weaddaflux correction ofgradients withatimestep( t) = ( t) to the first-order update. In general, the correction term [ ] ( t) l λp (l) λp (l) (46) x i+ i increases the accuracy of the l-th order update of q n i by one. 4. ADER for Multiple Space Dimensions We first consider the scalar two-dimensional linear advection equation t q +λ x q +λ y q =, (47) for q(x,y,t), where λ and λ are constant components of velocity. We are interested in un-split, explicit one-step finite volume schemes of the form qi,j n+ = qi,j n t [ ] f x i+,j f i,j t [ ] g y i,j+ g i,i, (48) where qi,j n is the integral average in a volume I i,j and f i+,j, g i,j+ are numerical fluxes. The ADER approach seeks a time average state at each interface to compute the corresponding numerical flux. Consider the truncated Taylor expansion in time about t = is q(x,y,τ) = m k= τ k k! (k) t q(x,y,). (49)

17 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 7 The time average of this between t = and t = t is given by which produces q = t t q = But the time derivatives are m { m k= k= ( t) k k! (k) t = ( λ () x λ () y ) k = A k ( k ξ } (k) t q(x,y,) dt, (5) ( t) k (k +)! (k) t q(x,y,). (5) ) ( λ ) ξ ( λ ) η (ξ) x (η) y, (5) which if used in (5) produce q = m k= ( t) k (k +)! A k ( ) k ( λ ) ξ ( λ ) η q (ξη) (x,y,), (53) ξ where A k = {(ξ,η) : ξ +η = k,ξ,η }. (54) As we are concerned with the average flux value over a cell boundary, at the cell inter-cell position x = x i+ for example, we have m q i+,j = k= t k ( λ )ξ ( λ )η q (ξη) (x (k +) ξ! η! y i+/,y,) dy, (55) A j k where q (ξη) denotes the mixed partial derivatives q (ξη) = (ξ) x (η) y q. (56) The numerical flux f i+,j at the inter-cell position x = x i+ by is then given f i+,j = f( q i+,j). (57) The three-dimensional case follows in an entirely analogous manner. The equation t q +λ x q +λ y q +λ 3 z q = (58)

18 8 E. F. TORO ET AL. for q(x,y,z,t), where λ, λ and λ 3 are constant components of velocity, is solved by the numerical scheme [ ] [ ] qi,j,k n+ = qn i,j,k t x f i+,j,k f i,j,k t y g i,j+,k g i,j,k [ t z h i,j,k+ h i,j,k The numerical flux f i+,j,k at the inter-cell position x = x i+ ] f ader i+ = f(qader,j,k i+. (59) is given by,j,k), (6) with m q ader i+,j,k = ( t) k ( λ )ξ ( λ )η ( λ 3 )ζ Q dy dz. (6) (k +) ξ! η! ζ! y z A j k k k= where the integrand Q involves mixed partial derivatives, namely and Q q (ξηζ) (x i+/,y,z,) = (ξ) x (η) y (ζ) z q(x i+/,y,z,) (6) A k = {(ξ,η,ζ) : ξ +η +ζ = k,ξ,η,ζ }. (63) Theinter-cell fluxesg i,j+,k and h i,j,k+ are entirely analogous to (6)-(6). We note here that the computation of the intercell fluxes in multiple space dimensions requires the solution of mixed derivative Riemann problems. Fortunately, Theorem for space derivatives in one space dimension extends to multiple space dimensions. The proof is simple and is therefore omitted here. We have presented the ADER approach for solving the linear advection equation with constant coefficients in one, two and three space dimensions. The schemes can thus be applied directly to linear systems with constant coefficients in one and multiple space dimensions. A pending problem is how to extend ADER to non-linear problems. In the next section we discuss some preliminary ideas. 5. ADER for Non-linear Scalar Equations Consider the non-linear scalar conservation law t q + x f(q) =, (64)

19 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 9 which in quasi-linear form reads where t q +λ(q) x f(q) =, (65) λ(q) = df(q) dq (66) is the characteristic speed, which in general is a function of the unknown q. Any extension of the ADER approach requires (i) replacement of the time derivatives (k) t q by space derivatives x (k) q via the differential equation and (ii) identification of appropriate evolution equations for the space derivatives. In the linear case step (i) is trivial, see equation (35); step (ii) is also trivial, as the evolution equation for space derivatives is the original linear advection equation, for all derivatives, see Theorem. Here we examine three possible approaches for extending ADER to non-linear problems such as (64). 5.. METHOD : SIMPLE LINEARISATION The approach relies on linearising the equations about the state q () in i+ (4) at the inter-cell in the flux expansion, as done for the Modified GRP scheme (Toro, 998). This term corresponds to the Godunov first-order upwind method and results from solving a non-linear Riemann problem. Then the characteristic speed is set to Then time derivatives are and the flux expansion is f ader i+ ˆλ () i+ = λ(q () ). (67) i+ (k) t q = ( ) kˆλk i+ x (k) q (68) t) k = f(q ader i+ ) ; q ader i+ = q () m ( ˆλ () i+ + i+ (k+)! k= q (k) i+. (69) The scheme is then identical to that for the linear advection equation. Preliminary results obtained by Millington et al. (Millington et al, 999)

20 E. F. TORO ET AL. and by Schwartzkopff (Schwartzkopff, 999) are available but we are not yet certain that this procedure will retain the sought accuracy. A systematic assessment of this method is a pending task. 5.. METHOD : RECURSIVE LINEARISATION A subtle modification of method results from a kind of recursive linearisation. Based on the time Taylor expansion (33) for the solution of the generalised Riemann problem we define a sequence of approximations in the following way Clearly Now define wave speeds s () = q () i+ s () = q () i+ s (l) = q () i+ +τ () + t q l k= τ k k! (k) t q (7) s (l) = s (l ) + τl l! (l) t q. (7) λ (k) i+ and a corresponding sequence of advection equations = λ(s (k) ) (7) t v +λ (k) i+ x v =, (73) where v x (k) q. To estimate the time derivative (k) t sequence to obtain we use the above ( k ) (k) t q = ( ) k λ (l) i+ x (k) q. (74) l= In order to evolve the spatial derivatives x (k) we use the latest available characteristic speed, that is the evolution equation (73) with speed λ (k ). i+ The final expression of the ADER flux is f ader i+ = f(q ader i+ ) ; q ader i+ = q () m ( t) k + i+ (k +)! k= ( k l= ) λ (l) q (k) i+ i+. (75)

21 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES We have not implemented this approach yet, but its simplicity looks very attractive METHOD 3: THE EXACT PROBLEM Another approach is to effectively solve the full problem. Here the k-th time derivatives are replaced by k-th space derivatives plus all extra terms involving lower order derivatives of q, the characteristic speed λ and mixed derivatives. Consider the exact problem (64)-(66). For a third-order method, for example, we need () t q = λ(q) () x q () t q = λ (q) x () q + dλ dq λ(q) ( () x q) (76) We also need to identify the evolution equations for the space derivatives. For a third-order method these are v () x q, () t v +λ () x v = v dλ dq v x () q, () t v +λ x () v = 3v( x () q)( dλ dq ) ( () x q) 3 ( d λ) dq (77) We are fortunate in that all the evolution equations have the same form; however they now have variable coefficients and source terms; these depend on the unknown of the problem and on lower-order derivatives. Applying the recursive procedure of method, the source terms are assumed to be known from the previous steps and thus can easily be evaluated. A systematic assessment of this approach is a pending task. 6. Numerical Experiments In this section we present some illustrative numerical results for the one and two dimensional linear advection equation with constant coefficients. First we solve the linear advection equation in (3) with speed λ =. The computational domain is [, ] is discretised into N equally-spaced cells of

22 E. F. TORO ET AL.. (i) ADER NTS=5 c=.8 N=5. (ii) ADER NTS=5 c=.8 N= (iii) ADER3 NTS=5 c=.8 N=5. (iv) ADER4 NTS=5 c=.8 N= (v) ADER5 NTS=5 c=.8 N=5. (vi) ADER NTS=5 c=.8 N= Figure 3. ADER computed results (symbol) for the linear advection equation after NTS = 5 time steps for smooth initial condition IC in (78) using N = 5 cells and CFL number C =.8. ADERk denotes the linear (fixed stencil) ADER scheme of k-th order of accuracy in space and time.

23 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 3 length x. Two initial conditions are considered, namely IC: Smooth q(x,) = e 8(x ) IC: Discontinuous q(x,) = Π. (x ) (78) The computations reported were performed on an sgi machine to double precision, on a mesh of N = 5 cells and a CFL coefficient c =.8. We applied periodic boundary conditions q(x, t) = q(x+, t) t. The solution was evolved by NTS = 5 time steps, which corresponds to a time of t = 8. units. Results are shown in Figs. 3 to 7, where the numerical computations, shown by symbols, are compared with the exact solution, shown by a full line. Fig. 3 shows computed results for the smooth initial condition in (78). The numerical results were obtained from linear ADER schemes (fixed stencil) of orders of accuracy,, 3, 4, 5 and ; see labels ADERk, where k corresponds to the order of accuracy. The numerical results, shown by symbols, are compared with the exact solution, shown by full line. ADER corresponds to the Godunov first-order upwind method. The scheme is monotone but is clearly too inaccurate, the extremum has been severely clipped to about 3% of its exact value. The second-order method is also very inaccurate, there is an obvious dispersive error and spurious oscillations are present, even for this problem with smooth solution. The third and fourth order methods give fairly accurate results, but it is only when we use the fifth order scheme that we compute a satisfactory result, at least for this output time and mesh. The th order method gives very accurate results. In Fig. 4 we show some results for the same problem comparing the non-linear ADER (adaptive stencil) schemes (ADER-LOS) with the corresponding ENO schemes used in conjunction with TVD Runge-Kutta ODE solvers. Figures on the left show ENO results and figures on the right show ADER-LOS results. Both second order non-linear methods give similar results and are both very inaccurate. Compare these second order results with the corresponding linear ADER result of Fig. 3. The spurious oscillations have been eliminated but the maximum has an error comparable to that of the first-order method. The third order schemes show a clear difference, with ADER-LOS giving a very accurate solution; compare ADER3-LOS with ADER3 of Fig. 3. The non-linear version of ADER has eliminated

24 4 E. F. TORO ET AL.. (vii) ENO NTS=5 c=.8 N=5. (x) ADER-LOS NTS=5 c=.8 N= (viii) ENO3 NTS=5 c=.8 N=5. (xi) ADER3-LOS NTS=5 c=.8 N= (ix) ENO4 NTS=5 c=.8 N=5. (xii) ADER4-LOS NTS=5 c=.8 N= Figure 4. Comparison of nonlinear ADER (denoted by ADER-LOS) with ENO schemes for smooth initial condition IC in (78) using N = 5 cells and CFL number C =.8. ENOk and ADERk denote the k-th order accurate ENO and ADER(non-linear) schemes.

25 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 5 the spurious oscillations of the linear ADER3 while preserving, to a good degree, the maximum value at the peak. The fourth-order ADER-LOS is again significantly more accurate than the fourth order ENO and again it preserves virtually the same value of the maximum computed by the linear ADER4 scheme, while eliminating the spurious oscillations. Figs. 5 and 6 show linear ADER results for the case of discontinuous initial condition in (78). As expected, all schemes of order greater than one produce spurious oscillations near discontinuities. Increasing the order of accuracy does not help significantly. Fig. 6 shows results from non-linear schemes, comparing ENO and ADER. All results are oscillation free; note that both second-order methods give similar but very inaccurate results. As seen for the smooth solution case, the third and fourth order ADER schemes are more accurate than the corresponding ENO schemes. WehaveimplementedtheweightedENO(WENO)ideainthecontext of ADER schemes to produce weighted ADER schemes (ADER-WLOS); here one takes a polynomial that is a convex combination of polynomials on all candidate stencils. For ENO schemes this procedure is known to increase the accuracy significantly. The same beneficial effect is observed in the ADER schemes, see results of Fig. 7. Again, the weighted ADER schemes are more accurate than the corresponding weighted ENO schemes. Millington (Millington, ) has carried out a systematic comparison between ADER and ENO schemes under a wide range of computational conditions and produced tabulated results of convergence rates and CPU times; these results will be published elsewhere. For the very simple problems solved here ADER gives superior results to ENO/WENO. The ADER schemes are also more efficient, by a factor of about 3, and we are confident that this computational cost difference will increase when solving more complex problems. As a second example we solved the two-dimensional linear advection equation (47) with λ = and λ = in a square domain [,] [,] with smooth initial condition q(x,y,) = e 8[(x ) +(y ) ] (79) and periodic boundary conditions.

26 6 E. F. TORO ET AL.. (i) ADER NTS=5 c=.8 N=5. (ii) ADER NTS=5 c=.8 N= (iii) ADER3 NTS=5 c=.8 N=5. (iv) ADER4 NTS=5 c=.8 N= (v) ADER5 NTS=5 c=.8 N=5. (vi) ADER NTS=5 c=.8 N= Figure 5. ADER computed results (symbol) for the linear advection equation after NTS = 5 time steps for discontinuous initial condition IC in (78) using N = 5 cells and CFL number C =.8. ADERk denotes the linear (fixed stencil) ADER scheme of k-th order of accuracy in space and time.

27 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 7 For the computations reported here we used a coarse regular mesh of 5 5 cells, a CFL coefficient c =.9 and evolved the solution for NTS = 3 time steps, which corresponds to an output time of t = 8 units. The results shown in Figs. 8 and 9 were obtained from the linear (oscillatory, fixed stencil interpolation) version of ADER. In the exact solution the maximum value is q max = and the minimum is q min =. The numerical values for maximum and minimum give a fair idea of the accuracy of the schemes. Fig. 8 shows the computed results for first, second and third order schemes (ADER, ADER and ADER3). The result from the first order scheme (ADER) can hardly be seen, the maximum has been very severely clipped to % of the correct value, q max =.. The results from the second and third order schemes (ADER and ADER3) can be seen but are still very inaccurate; they also exhibit spurious oscillations and their maxima are q max =.38 and q max =.568 respectively. Fig. 9 shows results obtained from the linear (oscillatory, fixed stencil interpolation) version of ADER for orders of accuracy 4,5 and. The respective maxima are q max =.75, q max =.86 and q max =.965. Of these, one may say that it is only the tenth-order result that is sufficiently accurate. This two-dimensional example on a coarse mesh shows quite dramatically that on the coarse meshes that are likely to be of practical use in real applications, low-order methods are simply too inaccurate to be used. 7. Conclusions and Further Developments The ADER approach for constructing non-oscillatory advection schemes of very high order of accuracy in space and time has been presented. The ADER formulation for linear advection with constant coefficients for one, two and three dimensions have been given in complete detail, along with some numerical results. Comparison with the state-of-the art ENO/WENO schemes has been made. The comparisons reveal that the ADER schemes are more accurate; they are also simpler and more efficient. Another attractive property of the ADER schemes, as compared with their closest

28 8 E. F. TORO ET AL.. (vii) ENO NTS=5 c=.8 N=5. (x) ADER-LOS NTS=5 c=.8 N= (viii) ENO3 NTS=5 c=.8 N=5. (xi) ADER3-LOS NTS=5 c=.8 N= (ix) ENO4 NTS=5 c=.8 N=5. (xii) ADER4-LOS NTS=5 c=.8 N= Figure 6. Comparison of nonlinear ADER (denoted by ADER-LOS) with ENO schemes for discontinuous initial condition IC in (78) using N = 5 cells and CFL number C =.8. ENOkand ADERkdenote the k-th order accurate ENO and ADER(non-linear) schemes.

29 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 9. (xiii) WENO3 NTS=5 c=.8 N=5. (xiv) ADER3-WLOS NTS=5 c=.8 N= (xiii) WENO3 NTS=5 c=.8 N=5. (xiv) ADER3-WLOS NTS=5 c=.8 N= Figure 7. Comparison of nonlinear Weighted ADER (denoted by ADER-WLOS) with Weighted ENO schemes for smooth initial condition IC in (78) using N = 5 cells and CFL number C =.8. ENOk and ADERk denote the k-th order accurate ENO and ADER (non-linear) schemes. competitors, the ENO/WENO schemes, is that for ADER we have not yet found accuracy barriers, at least for linear systems with constant coefficients. The ENO/WENO schemes rely on TVD solvers for the associated ordinary differential equations in time; so far the most accurate TVD ODE solver developed for use with ENO/WENO schemes is of fifth order of accuracy. As they stand, the ADER schemes can be extended to linear systems with constant coefficients. Schwartzkopff (Schwartzkopff, 999) has already applied the ADER approach to the linearised Euler equations of gas dynamics with very encouraging results. Munz and Schneider (Munz

30 3 E. F. TORO ET AL. Ader Ader Ader Figure 8. Two-dimensional ADER results for linear advection equation with constant coefficients λ = λ =. Results shown are for the linear (fixed stencil) ADER schemes of first, second and third order accuracy.

31 TOWARDS VERY HIGH ORDER GODUNOV SCHEMES 3 Ader Ader Ader Figure 9. Two-dimensional ADER results for linear advection equation with constant coefficients λ = λ =. Results shown are for the linear (fixed stencil) ADER schemes of fourth, fifth and tenth order accuracy.

32 3 E. F. TORO ET AL. and R. Schneider, ) have extended the ADER approach to unstructured meshes to solve the two-dimensional linearised Maxwell equations. In spite of these two-dimensional applications the practical experience on implementation issues in still very limited; there is so far no experience on three-dimensional implementations. Preliminary results on the application of centred fluxes,ratherthan upwind,inthecontext ofader show that for the low order range upwind schemes are more accurate but for high order of accuracy the results are virtually indistinguishable. This is an area that requires more work. It may well be that one can use approximate Riemann solvers for the first few terms in the flux expansion, say two, and centred schemes for the remaining terms. The next important development of the ADER approach is its extension to non-linear problems. Here we have presented three potential approaches, two of them relying heavily on local linearisations. Limited practical experience is available with one of the approaches, the simplest, but it is so far uncertain as to whether such simple approach is capable of retaining the expected high accuracy We are confident that the other two approaches or some combination of the two will be productive, but this is speculative at this stage. Another possibility is the use of the relaxation approach of Shi and (Shi and X. Zhouping, 995). Schwartzkopff and Schroll (private communication) are already working in this direction. The ADER schemes for linear systems with constant coefficients in one and multiple space dimensions are so far reasonably well understood, although one would like to be able to prove rigourously the stability limit of the schemes, for example. The indications so far are that the schemes have stability unity for all orders in one and multiple space dimensions. Analysis of ADER schemes for non-linear schemes, if successful, is bound to be simpler than for multi-level schemes.

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