Eleuterio F. Toro

Restoration of the contact surface in the HLL-Riemann solver

The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.

Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems

Experimental and numerical results concerning the flow induced by the break of a dam on a dry bed are presented. The numerical technique consists of a shock-capturing method of the Godunov type. A physical laboratory model has been employed to infer properties and validity of the numerical solution. Attention is also given to the applicability of the mathematical model, based on the shallow water equations, to this class of problems.

ADER: Arbitrary High Order Godunov Approach

This paper concerns the construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws. The schemes result from extending the ADER approach, which is related to the ENO/WENO methodology. Our 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; the schemes have optimal stability condition. To compute the intercell flux in one space dimension we solve a generalised Riemann problem by reducing it to the solution a sequence of m conventional Riemann problems for the kth spatial derivatives of the solution, with k=0, 1,..., m−1, where m is arbitrary and is the order of the accuracy of the resulting scheme. We provide numerical examples using schemes of up to fifth order of accuracy in both time and space.

Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

Abstract In this article we present a quadrature-free essentially non-oscillatory finite volume scheme of arbitrary high order of accuracy both in space and time for solving nonlinear hyperbolic systems on unstructured meshes in two and three space dimensions. For high order spatial discretization, a WENO reconstruction technique provides the reconstruction polynomials in terms of a hierarchical orthogonal polynomial basis over a reference element. The Cauchy–Kovalewski procedure applied to the reconstructed data yields for each element a space–time Taylor series for the evolution of the state and the physical fluxes. This Taylor series is then inserted into a special numerical flux across the element interfaces and is subsequently integrated analytically in space and time. Thus, the Cauchy–Kovalewski procedure provides a natural, direct and cost-efficient way to obtain a quadrature-free formulation, avoiding the expensive numerical quadrature arising usually for high order finite volume schemes in three space dimensions. We show numerical convergence results up to sixth order of accuracy in space and time for the compressible Euler equations on triangular and tetrahedral meshes in two and three space dimensions. Furthermore, various two- and three-dimensional test problems with smooth and discontinuous solutions are computed to validate the approach and to underline the non-oscillatory shock-capturing properties of the method.

A weighted average flux method for hyperbolic conservation laws

A numerical technique, called a ‘weighted average flux’ (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving ‘generalized’ Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

Solution of the generalized Riemann problem for advection–reaction equations

We present a method for solving the generalized Riemann problem for partial differential equations of the advection–reaction type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy-Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy-Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space-time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space-time integration over each control volume, using the local space-time DG solutions at the Gaussian integration points for the intercell fluxes and for the space-time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187-210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235-277] and the full compressible Euler equations with stiff friction source terms.

Towards Very High Order Godunov Schemes

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 10-th order accuracy are presented.

A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems

We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math. Comput. 38:339–374, 1982) to a certain class of hyperbolic systems in non-conservative form, in particular to shallow-water-type and multi-phase flow models. To this end we apply the formalism of path-conservative schemes introduced by Parés (SIAM J. Numer. Anal. 44:300–321, 2006) and Castro et al. (Math. Comput. 75:1103–1134, 2006). For the sake of generality and simplicity, we suggest to compute the inherent path integral numerically using a Gaussian quadrature rule of sufficient accuracy. Published path-conservative schemes to date are based on either the Roe upwind method or on centered approaches. In comparison to these, the proposed new path-conservative Osher-type scheme has several advantages. First, it does not need an entropy fix, in contrast to Roe-type path-conservative schemes. Second, our proposed non-conservative Osher scheme is very simple to implement and nonetheless constitutes a complete Riemann solver in the sense that it attributes a different numerical viscosity to each characteristic field present in the relevant Riemann problem; this is in contrast to centered methods or incomplete Riemann solvers that usually neglect intermediate characteristic fields, hence leading to excessive numerical diffusion. Finally, the interface jump term is differentiable with respect to its arguments, which is useful for steady-state computations in implicit schemes. We also indicate how to extend the method to general unstructured meshes in multiple space dimensions. We show applications of the first order version of the proposed path-conservative Osher-type scheme to the shallow water equations with variable bottom topography and to the two-fluid debris flow model of Pitman & Le. Then, we apply the higher-order multi-dimensional version of the method to the Baer–Nunziato model of compressible multi-phase flow. We also clearly emphasize the limitations of our approach in a special chapter at the end of this article.

Derivative Riemann solvers for systems of conservation laws and ADER methods

In this paper, we first briefly review the semi-analytical method [E.F. Toro, V.A. Titarev, Solution of the generalized Riemann problem for advection-reaction equations, Proc. Roy. Soc. London 458 (2018) (2002) 271-281] for solving the derivative Riemann problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it 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 the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes.