Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
A numerical method for viscous perturbations of hyperbolic conservation laws
SIAM Journal on Numerical Analysis
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Journal of Computational Physics
The Travelling Wave Scheme for The Navier--Stokes Equations
SIAM Journal on Numerical Analysis
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Derivative Riemann solvers for systems of conservation laws and ADER methods
Journal of Computational Physics
Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes
Journal of Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
ADER finite volume schemes for nonlinear reaction--diffusion equations
Applied Numerical Mathematics
Polymorphic nodal elements and their application in discontinuous Galerkin methods
Journal of Computational Physics
Journal of Scientific Computing
Stable Interface Conditions for Discontinuous Galerkin Approximations of Navier-Stokes Equations
Journal of Scientific Computing
Discretisation of diffusive fluxes on hybrid grids
Journal of Computational Physics
Journal of Computational Physics
The Elastoplast Discontinuous Galerkin (EDG) method for the Navier-Stokes equations
Journal of Computational Physics
Recovery of normal derivatives from the piecewise L2 projection
Journal of Computational Physics
A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations
Journal of Scientific Computing
Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations
Journal of Computational Physics
WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flows
Journal of Computational Physics
Journal of Computational Physics
High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows
Journal of Computational Physics
Hi-index | 31.50 |
In this paper, we consider numerical approximations of diffusion terms for finite volume as well as discontinuous Galerkin schemes. Both classes of numerical schemes are quite successful for advection equations capturing strong gradients or even discontinuities, because they allow their approximate solutions to be discontinuous at the grid cell interfaces. But, this property may lead to inconsistencies with a proper definition of a diffusion flux. Starting with the finite volume formulation, we propose a numerical diffusion flux which is based on the exact solution of the diffusion equation with piecewise polynomial initial data. This flux may also be used by discontinuous Galerkin schemes and gives a physical motivation for the Symmetric Interior Penalty discontinuous Galerkin scheme. The flux proposed leads to a one-step finite volume or discontinuous Galerkin scheme for diffusion, which is arbitrary order accurate simultaneously in space and time. This strategy is extended to define suitable numerical fluxes for nonlinear diffusion problems.