Nonconformal schemes of the finite-element method for nonlinear hyperbolic conservation laws

2011 ◽  
Vol 47 (8) ◽  
pp. 1197-1209 ◽  
Author(s):  
E. M. Fedotov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
The Finite Element Method ◽  
Hyperbolic Conservation ◽  
Nonlinear Hyperbolic Conservation Laws ◽  
Element Method

scholarly journals Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Aziz Madrane ◽  
Fayssal Benkhaldoun
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Conservation Laws ◽  
Discontinuous Galerkin ◽  
Hyperbolic Conservation Laws ◽  
Numerical Dissipation ◽  
Hyperbolic Conservation ◽  
High Mach ◽  
Entropy Stable ◽  
Element Method

Abstract We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.

Dual System Least-Squares Finite Element Method for a Hyperbolic Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Eunjung Lee ◽  
Hyesun Na
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Hyperbolic Conservation Laws ◽  
Dual System ◽  
Hyperbolic Problem ◽  
Newton’S Iterative Method ◽  
Hyperbolic Conservation ◽  
The One ◽  
Element Method

Abstract This study investigates the dual system least-squares finite element method, namely the LL∗ method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL∗ method and Newton’s iterative method. In addition, the inclusion of a stabilizing term in the discrete LL∗ minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.

CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS

1995 ◽  
Vol 05 (03) ◽  
pp. 367-386 ◽  
Author(s):  
J. JAFFRE ◽  
C. JOHNSON ◽  
A. SZEPESSY
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Conservation Laws ◽  
Discontinuous Galerkin ◽  
Hyperbolic Conservation Laws ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Systems Of Conservation Laws ◽  
Element Method

We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

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