Abstract: We present new truly multidimensional schemes of higher order within the framework of finite volume evolution Galerkin (FVEG) methods for systems of nonlinear hyperbolic conservation laws. These methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system, such that all of the infinitely many directions of wave propagation are taken into account. Following our previous results for the wave equation system approximate evolution operators for the linearised Euler equations are derived. The integrals along the Mach cone and along the cell interfaces are evaluated exactly as well as by means of numerical quadratures. The influence of these numerical quadratures will be discussed. Second order resolution is obtained using a conservative piecewise bilinear recovery and the midpoint rule approximation for time integration. We prove error estimates for the finite volume evolution Galerkin scheme for linear systems with constant coefficinets. Several numerical experiments for the nonlinear Euler equations, which confirm the accuracy and good multidimensional behaviour of the FVEG schemes, are presented as well.
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