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   Table of Diffusion Schemes
Below, various approaches to the construction of numerical schemes for the diffusion equation:


where is a constant diffusion coefficient, are categorized into two groups: second-order and higher-order methods. The approaches are categorized in term of the key framework in constructing a diffusion scheme. Acronyms in the parenthesis indicate what method the technique is applicable to, designed for, and/or actually used in.

- Only those that solve the diffusion equation are considered.
- The focus is on the construction method for diffusion schemes.
- Some of the resulting schemes may turn out to be equivalent.
- Higher-order methods can be a second-order method.
- Application of some schemes to unstructured grids may be difficult.
- Extensions to the Navier-Stokes equations may be difficult in some schemes.
- Some schemes are based on more than one approach.
- Only the following methods are considered at the moment.
FV:  Finite Volume
FD:  Finite Difference
RD:  Residual Distribution
FE:  Finite Element
DG:  Discontinuous Galerkin
SV:  Spectral Volume
SD:  Spectral Difference
Approach Second-Order Higher-Order
Derivative Evaluation:

[ Differential Form of Diffusion ]
Central Difference (FD)
Gridless/Meshless Schemes
Compact Schemes - Lele, RBC (FD)
Reconstruction - WENO (FD)
Interface Gradient (or Flux):

[ Integral Form of Diffusion ]
Average LSQ Gradient (FV)
Green-Gauss (FV)
Braaten-Connell (FV)
Central Difference (FV)
Reconstruction - WENO (FV)
Interior-Penalty (DG,SV,SD)
Recovery Schemes (DG)
Flux Reconstruction Schemes (DG,SV,SD)
Direct DG - DDG (DG)
Riemann solver for diffusion (FV,DG)
Weak/Variational Formulation:

[ See FEM textbooks ]
Galerkin (FE,FV,RD) Petrov-Galerkin (RD)
P2 Galerkin (FE,RD)
Mixed Formulation:

[ See FEM textbooks ]
Mixed Finite-Element (FE)
LSQ Finite-Element - FOSLS (FE)
Bassi-Rebay and LDG (DG,SV,SD)
Nishikawa-Roe (RD)
Utilization of Alternative Model:
Hyperbolic model, Relaxation model, Kinetic model, etc.

[ Note: Not to solve them, but to derive a diffusion scheme. ]
Hyperbolic Model (All Methods)
Kinetic method (FV,RD)
Hyperbolic Model (All Methods)
Kinetic method (DG)
Hyperbolic System Approach:

[ Nishikawa, JCP2007 ]
Upwind Schemes (All Methods) Upwind Schemes (All Methods; steady)

LSQ: Least-Squares,    WENO: Weighted-Essentially Non-Oscillatory schemes,    RBC: Residual-Based Compact schemes,    DDG: Direct Discontinuous Galerkin method,    LDG: Local Discontinuous Galerkin method.   
To be added: CDG method
More details, discussions, more schemes, and/or other categorizations may be added later.
Comments and suggestions are always welcome! [ E-mail ]

by Hiroaki Nishikawa, December 16, 2010.
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Copyright 1994- by Hiroaki Nishikawa. All rights reserved.