Derivation of Strictly Stable High Order Difference Approximations for Variable-Coefficient PDE

Authors:
Katharina Kormann;Martin Kronbichler;Bernhard Müller
Affiliations:
Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden;Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden;Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Venue:
Journal of Scientific Computing
Year:
2012

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Abstract

A new way of deriving strictly stable high order difference operators for partial differential equations (PDE) is demonstrated for the 1D convection diffusion equation with variable coefficients. The derivation is based on a diffusion term in conservative, i.e. self-adjoint, form. Fourth order accurate difference operators are constructed by mass lumping Galerkin finite element methods so that an explicit method is achieved. The analysis of the operators is confirmed by numerical tests. The operators can be extended to multi dimensions, as we demonstrate for a 2D example. The discretizations are also relevant for the Navier---Stokes equations and other initial boundary value problems that involve up to second derivatives with variable coefficients.