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Preface to the special issue “Recent progress in numerical methods for nonlinear time-dependent flow & transport problems”

Authors:
  • Federal Researcher Center "Computer Science and Control" of Russian academy of sciences
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Abstract

This special issue is dedicated to recent developments concerning modern numerical methods for time dependent partial differential equations that can be used to simulate complex nonlinear flow and transport processes and tries to cover a wide spectrum of different applications and numerical approaches. Most of the models under consideration here are nonlinear hyperbolic conservation laws, which are a very powerful mathematical tool to describe a large set of problems in science and engineering. Hyperbolic PDE are based on first principles, such as mass, momentum and total energy conservation as well as the second law of thermodynamics. These principles are universally valid and have been successfully used for the simulation of a very wide class of problems, ranging from astrophysics over geophysics to complex flows in engineering and computational biology. Most of the seemingly different applications listed above have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher-order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise from the nonlinearities that allow the formation of non-smooth solutions such as shock waves, material interfaces or other discontinuities, but nonlinear conservation laws can simultaneously also contain smooth features like sound waves and vortex flows. Even nowadays the key challenge for the construction of successful numerical methods for nonlinear hyperbolic PDE is therefore to capture at the same time shock waves and discontinuities robustly without producing spurious oscillations and exhibit little numerical dissipation and dispersion errors in smooth regions of the solution. For this reason, the development of robust and accurate numerical methods for nonlinear flow and transport problems remains a great challenge even after several decades of successful research, which started back in the 1940s and 1950s with the pioneering works of von Neumann [1] and Godunov [2]. The present special issue is published on the occasion of the 70th birthday of Prof. Eleuterio F. Toro and in honor of his outstanding contributions to the fields of applied mathematics, scientific computing and computational physics with his well-known numerical methods for the solution of hyperbolic conservation laws and their application to computational fluid dynamics, computational physics and computational biology.

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We present first- and higher-order non-oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non-conservative) form. Non-conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non-conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second-order non-oscillatory methods. We have used the MUSCL-Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher-order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non-conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind-based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd.
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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.
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The equations of hydrodynamics are modified by the inclusion of additional terms which greatly simplify the procedures needed for stepwise numerical solution of the equations in problems involving shocks. The quantitative influence of these terms can be made as small as one wishes by choice of a sufficiently fine mesh for the numerical integrations. A set of difference equations suitable for the numerical work is given, and the condition that must be satisfied to insure their stabilty is derived.
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We first construct an approximate Riemann solver of the HLLC-type for the Baer–Nunziato equations of compressible two-phase flow for the “subsonic” wave configuration. The solver is fully nonlinear. It is also complete, that is, it contains all the characteristic fields present in the exact solution of the Riemann problem. In particular, stationary contact waves are resolved exactly. We then implement and test a new upwind variant of the path-conservative approach; such schemes are suitable for solving numerically nonconservative systems. Finally, we use locally the new HLLC solver for the Baer–Nunziato equations in the framework of finite volume, discontinuous Galerkin finite element and path-conservative schemes. We systematically assess the solver on a series of carefully chosen test problems.
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In this paper we present the exact solution of the Riemann problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave. We provide examples of possible wave patterns. Numerical solution of a first-order dissipative scheme as well as an implementation of our Riemann solver in the second-order upwind method are compared with the proposed exact Riemann problem solution. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.
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This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.
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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.
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A unified, Riemann-problem-based extension of the Warming–Beam and Lax- Wendroff schemes for nonlinear systems of hyperbolic conservation laws is presented. This is achieved by employing the weighted average flux (WAF) approach with the numerical flux given by a space–time integral average of the solution of local Riemann problems. The local wave structure dictates switching between schemes automatically with no need for special conservative switching operators. The resulting Godunov-type scheme has CFL-number 2 stability restriction and when used in conjunction with Riemann-solver adaption procedures is significantly more efficient than Riemann-problem-based methods currently in use.