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Adaptive backward difference formula–Discontinuous Galerkin finite element method for the solution of conservation laws
Author(s) -
Dolejší V.,
Kůs P.
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2143
Subject(s) - discontinuous galerkin method , conservation law , mathematics , discretization , finite element method , inviscid flow , galerkin method , piecewise , linearization , partial differential equation , order of accuracy , piecewise linear function , mathematical analysis , nonlinear system , method of characteristics , classical mechanics , physics , quantum mechanics , thermodynamics
We deal with the numerical solution of the system of conservation laws. Although this approach has been proposed for a simulation of inviscid compressible flow, it can be straightforwardly applied to more general problems. We carried out the space semi‐discretization by the discontinuous Galerkin finite element (DGFE) method, which is based on a piecewise polynomial discontinuous approximation. The resulting system of ordinary differential equations is discretized by the backward difference formula (BDF). A suitable linearization of the physical fluxes leads to a scheme that is practically unconditionally stable and has a higher order of accuracy with respect to the space and time coordinates and we solve a linear algebraic system at each time level. Moreover, we develop an adaptive technique for a choice of the length of the time step that is based on the use of two BDFs of the same order of accuracy. We call the resulting scheme the ABDF–DGFE (adaptive BDF–DGFE)method. Finally, the efficiency of the presented adaptive strategy is documented by a set of numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.