Core Concepts
Dafermos' entropy rate criterion enhances stability in DG methods on unstructured grids.
Abstract
The article extends the approach to multidimensional systems of conservation laws using Discontinuous Galerkin methods on triangular grids. Special attention is given to entropy dissipation from boundaries, resulting in schemes free of tunable viscosity parameters. Theoretical background on hyperbolic systems and balance laws is discussed, along with the application of Dafermos' entropy rate criterion for numerical approximations. The implementation of filters for dissipating entropy and accuracy tests are also presented.
Introduction:
Solvers for hyperbolic systems are crucial in computational fluid dynamics.
Existence and uniqueness theory for associated equations remain open.
Theory:
Multidimensional DG schemes are high-order generalizations of Finite Volume methods.
Discontinuous Galerkin method allows discontinuities between cells.
Implementation:
Correction direction υ is calculated using a dissipation generator G.
Numerical Tests:
Accuracy test results show high-order accuracy and mesh independence for polynomial degrees 1 and 3.
Sedov Blast Wave:
Simulation of Sedov's blast wave problem shows non-symmetric solutions due to contact discontinuity instability.
Stats
Schemes satisfying additional criterion constructed [21, 22, 23].
Dissipation speed bounded by conservative filter [28].
Sedov blast wave initial conditions: ρin = 1.0, vin = 0, pin = 1.0, ρout = 0.125, vout = 0, pout = 0.1.