A hybridized discontinuous Petrov-Galerkin scheme for scalar conservation laws

SUMMARY We present a hybridized discontinuous Petrov–Galerkin (HDPG) method for the numerical solution of steady and time‐dependent scalar conservation laws. The method combines a hybridization technique with a local Petrov–Galerkin approach in which the test functions are computed to maximize the i...

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Bibliographic Details
Published in:International journal for numerical methods in engineering Vol. 91; no. 9; pp. 950 - 970
Main Authors: Moro, D., Nguyen, N. C., Peraire, J.
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 31.08.2012
Wiley
Subjects:
Classical transport
Exact sciences and technology
hybridized discontinuous Galerkin
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Nonlinear algebraic and transcendental equations
nonlinear conservation laws
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Petrov-Galerkin
Physics
Sciences and techniques of general use
Statistical physics, thermodynamics, and nonlinear dynamical systems
Transport processes
Performance evaluation
Discontinuity
Costs
Constraint
nonlinear conservation laws
Non linear system
Modeling
Newton Raphson method
Trace
Galerkin-Petrov method
Computation time
Conservation law
Linear system
Petrov-Galerkin
hybridized discontinuous Galerkin
Galerkin method
Non linear effect
Numerical convergence
ISSN:0029-5981, 1097-0207
Online Access:Get full text
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Summary:SUMMARY We present a hybridized discontinuous Petrov–Galerkin (HDPG) method for the numerical solution of steady and time‐dependent scalar conservation laws. The method combines a hybridization technique with a local Petrov–Galerkin approach in which the test functions are computed to maximize the inf‐sup condition. Since the Petrov–Galerkin approach does not guarantee a conservative solution, we propose to enforce this explicitly by introducing a constraint into the local Petrov–Galerkin problem. When the resulting nonlinear system is solved using the Newton–Raphson procedure, the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace. This results in a significant reduction in memory storage and computation time for the solution of the matrix system, albeit at the cost of solving the local Petrov–Galerkin problems. However, these local problems are independent of each other and thus perfectly scalable. We present several numerical examples to assess the performance of the proposed method. The results show that the HDPG method outperforms the hybridizable discontinuous Galerkin method for problems involving discontinuities. Moreover, for the test case proposed by Peterson, the HDPG method provides optimal convergence of order k + 1. Copyright © 2012 John Wiley & Sons, Ltd.
Bibliography:ark:/67375/WNG-HHC78RL8-B
ArticleID:NME4300
istex:6EBD8D85C95061C69BA386FEF1BA9FE15F28C063
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.4300