DIMEX Runge-Kutta finite volume methods for multidimensional hyperbolic systems

Authors:
Enrico Bertolazzi;Gianmarco Manzini
Affiliations:
Dipartimento di Ingegneria Meccanica e Strutturale, Universití di Trento, via Mesiano 77, I-38050 Trento, Italy;Istituto di Matematica Applicata e Tecnologie Informatiche, IMATI-CNR, via Ferrata 1, I-27100 Pavia, Italy
Venue:
Mathematics and Computers in Simulation
Year:
2007

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Abstract

We propose a class of finite volume methods for the discretization of time-dependent multidimensional hyperbolic systems in divergence form on unstructured grids. We discretize the divergence of the flux function by a cell-centered finite volume method whose spatial accuracy is provided by including into the scheme non-oscillatory piecewise polynomial reconstructions. We assume that the numerical flux function can be decomposed in a convective term and a non-convective term. The convective term, which may be source of numerical stiffness in high-speed flow regions, is treated implicitly, while the non-convective term is always discretized explicitly. To this purpose, we use the diagonally implicit-explicit Runge-Kutta (DIMEX-RK) time-marching formulation. We analyze the structural properties of the matrix operators that result from coupling finite volumes and DIMEX-RK time-stepping schemes by using M-matrix theory. Finally, we show the behavior of these methods by some numerical examples.