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Numerical solutions of systems with ( p , δ )‐structure using local discontinuous Galerkin finite element methods
Author(s) -
Kröner Dietmar,
Růžička Michael,
Toulopoulos Ioannis
Publication year - 2014
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.3955
Subject(s) - mathematics , discretization , nonlinear system , discontinuous galerkin method , galerkin method , runge–kutta methods , finite element method , numerical analysis , algebraic equation , mathematical analysis , physics , quantum mechanics , thermodynamics
SUMMARY In this paper, we present LDG methods for systems with ( p , δ )‐structure. The unknown gradient and the nonlinear diffusivity function are introduced as auxiliary variables and the original ( p , δ ) system is decomposed into a first‐order system. Every equation of the produced first‐order system is discretized in the discontinuous Galerkin framework, where two different nonlinear viscous numerical fluxes are implemented. An a priori bound for a simplified problem is derived. The ODE system resulting from the LDG discretization is solved by diagonal implicit Runge–Kutta methods. The nonlinear system of algebraic equations with unknowns the intermediate solutions of the Runge–Kutta cycle is solved using Newton and Picard iterative methodology. The performance of the two nonlinear solvers is compared with simple test problems. Numerical tests concerning problems with exact solutions are performed in order to validate the theoretical spatial accuracy of the proposed method. Further, more realistic numerical examples are solved in domains with non‐smooth boundary to test the efficiency of the method. Copyright © 2014 John Wiley & Sons, Ltd.