Finite Volume Methods for Scalar Conservation Laws on Time Dependent Meshes

Finite volume method is a method of choice for hyperbolic systems of conservation laws such as the Euler equations of gas dynamics. FVM is often combined with mesh adaption techniques. Since rigorous treatment of hyperbolic systems is far beyond current state of research, we use initial-boundary value problem for scalar conservation law as a model case. We estabilish basic form of an algorithm which couples finite volume time step with recomputation of solution to a new mesh on each time level. We discuss ideas which are related to recomputation procedures such as maximum principle, BV stability, geometric mass conservation and compatibility with entropy inequailty. We propose a way to incorporate the recomputation procedure in proof of convergence of the algorithm. Eventually, we set a plan for future work in the area. Introduction Hyperbolic systems of conservation laws have important applications in computational fluid dynamics. The basic example is the system of the Euler equations of gas dynamics, see Feistauer et al. [2003], Godlewski and Raviart [1996]. Although there was some progress concerning special cases of hyperbolic systems, fundamental theoretical questions concerning weak formulation and well-posedness, especially in multiple spatial dimensions, were still not answered. Since there are no analytical tools, theory of the basic numerical method for hyperbolic systems, the finite volume method, remains incomplete, too. Theory of scalar conservation laws and corresponding numerical methods are much more developed, see e.g. Málek et al. [1996]. Scalar problems have interesting properties which can be exploited in analysis by numerical methods, such as maximum principle. The main theoretical tool is the entropy inequality and the method of doubling variables of Kruzhkov. Error of various approximations to weak entropy solutions can be estimated with help of residuals in entropy inequality, see Bouchut and Perthame [1998]. This approach was successfully used to obtain convergence results and error estimates for finite volume method, see e.g. Eymard et al. [2000]. In typical case, solution of conservation law is not continuous. It is very desirable for computationally efficient numerical method to involve some mesh adaption techniques. For evolution problems, mesh adaptivity require that the numerical method works with different meshes on different time levels. The basic form of FVM time step involves only one mesh and the solution must be recomputed or interpolated to the mesh of next time step by some artificial procedure. Such combined method was proposed by Tang and Tang [2003]. Their recomputation procedure is a conservative update with numerical fluxes, which resemble the finite volume method itself. See also Felcman and Kubera [2006] for numerical experiments on similar adaptive method. Tang and Tang [2003] proved convergence to the weak solution in the case of one space dimension. Their recomputation procedure is also stable in L∞ and in BV . Although they do not use the concept of entropy weak solution, the similarity of the recomputation procedure with finite volume formulation leads to some discrete entropy inequality. The aim of our present and future work is to extend these results to multiple space dimensions, to problems with boundary conditions and to other recomputation procedures. The WDS'07 Proceedings of Contributed Papers, Part I, 215–220, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS