scholarly journals Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity

2019 ◽  
Vol 391 ◽  
pp. 14-36
Author(s):  
Hui Yu ◽  
Hailiang Liu
Keyword(s):  
Maximum Principle ◽  
Third Order ◽  
Convection Diffusion ◽  
Diffusion Problems

scholarly journals An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes

2017 ◽  
Vol 27 (03) ◽  
pp. 525-548 ◽  
Author(s):  
Gabriel R. Barrenechea ◽  
Volker John ◽  
Petr Knobloch
Keyword(s):  
Maximum Principle ◽  
Discrete Maximum Principle ◽  
Flux Correction ◽  
Finite Element Scheme ◽  
Convection Diffusion ◽  
Flux Limiter ◽  
Correction Scheme ◽  
Definition Of ◽  
General Meshes ◽  
Diffusion Problems

This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection–diffusion problems confirm the theory.

Error estimates of the third order runge-kutta alternating evolution discontinuous galerkin method for convection-diffusion problems

2018 ◽  
Vol 52 (5) ◽  
pp. 1709-1732
Author(s):  
Hailiang Liu ◽  
Hairui Wen
Keyword(s):  
Stability Analysis ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Energy Estimates ◽  
Third Order ◽  
Convection Diffusion ◽  
Temporal Discretization ◽  
Runge Kutta ◽  
The Stability ◽  
Diffusion Problems

In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition c0τ ≤ ε ≤ c1h2. Here ε is the method parameter, and h is the maximum spatial grid size. We further obtain the optimal L2 error of order O(τ3 + hk+1). Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

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