Colloquia
Michael Dumbser
CAM COLLOQUIUM
“Quadrature-Free WENO Finite-Volume Schemes for Nonlinear Hyperbolic Systems on Unstructured Triangular and Tetrahedral Meshes in Two and Three Dimensions”
Michael Dumbser
Laboratory of Applied Mathematics, Faculty of Engineering
University of Trento, Italy
Wednesday, September 26, 2007
4:00PM in 231 Hayes-Healy Center
*TEA – 3:30PM in Math Lounge – 257 Hayes-Healy*
Abstract:
In our talk we present a quadrature-free weighted essentially non-oscillatory (WENO) finite volume scheme of arbitrary high order of accuracy both in space and time for solving time-dependent nonlinear hyperbolic systems on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively.
For high order spatial discretization, a WENO reconstruction technique provides the reconstruction polynomials in terms of a hierarchical orthogonal polynomial basis over a reference element. To ensure non-oscillatory behaviour also for nonlinear systems, characteristic reconstruction is used.
The Cauchy-Kovalewski procedure applied to the reconstructed data yields for each element a space-time Taylor series for the evolution of the state and the physical fluxes. This Taylor series is then inserted into a new numerical flux which is a function of four arguments and which can be subsequently integrated analytically in space and time. Thus, the Cauchy-Kovalewski procedure provides a natural, direct and cost-efficient way to obtain a quadrature-free formulation, avoiding the expensive numerical quadrature arising usually for high order finite volume schemes in three space dimensions. We show numerical convergence results up to sixth order of accuracy in space and time for the compressible Euler equations of gas dynamics on triangular and tetrahedral meshes in two and three space dimensions. Furthermore, various unsteady two- and three-dimensional flow problems with smooth and discontinuous solutions are computed to validate the accuracy of the approach and to underline the non-oscillatory shock-capturing properties of the method. High order accurate curvilinear boundary treatment within our high-order finite volume framework is also discussed.
To our knowledge, this is the first high order accurate quadrature-free WENO finite volume scheme ever presented for nonlinear hyperbolic systems on unstructured three-dimensional tetrahedral meshes. As a final outlook to future applications, first results for the compressible unsteady Navier-Stokes equations on unstructured triangular meshes are shown.
