Convergence of a Finite Volume Extension of the Nessyahu–tadmor Scheme on Unstructured Grids for a Two-dimensional Linear Hyperbolic Equation∗

Abstract. The nonoscillatory central difference scheme of Nessyahu and Tadmor is a Godunovtype scheme for one-dimensional hyperbolic conservation laws in which the resolution of Riemann problems at the cell interfaces is bypassed thanks to the use of the staggered Lax–Friedrichs scheme. Piecewise linear MUSCL-type (monotonic upstream-centered scheme for conservation laws) cell interpolants and slope limiters lead to an oscillation-free second-order resolution. Convergence to the entropic solution was proved in the scalar case. After extending the scheme to a two-step finite volume method for two-dimensional hyperbolic conservation laws on unstructured grids, we present here a proof of convergence to a weak solution in the case of the linear scalar hyperbolic equation ut + div(~ V u) = 0. Since the scheme is Riemann solver–free, it provides a truly multidimensional approach to the numerical approximation of compressible flows, with a firm mathematical basis. Numerical experiments show the feasibility and high accuracy of the method.