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.
Note:
- 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.
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| FV: | Finite Volume |
| FD: | Finite Difference |
| RD: | Residual Distribution |
| FE: | Finite Element |
| DG: | Discontinuous Galerkin |
| SV: | Spectral Volume |
| SD: | Spectral Difference |
Approach
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Second-Order
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Higher-Order
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Derivative Evaluation:
[ Differential Form of Diffusion ]
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Central Difference (FD)
Gridless/Meshless Schemes
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Compact Schemes -
Lele,
RBC (FD)
Reconstruction - WENO (FD)
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Interface Gradient (or Flux):
[ Integral Form of Diffusion ]
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Average LSQ Gradient (FV)
Green-Gauss (FV)
Braaten-Connell (FV)
Central Difference (FV)
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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)
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Weak/Variational Formulation:
[ See FEM textbooks ]
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Galerkin (FE,FV,RD)
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Petrov-Galerkin (RD)
P2 Galerkin (FE,RD)
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Mixed Formulation:
[ See FEM textbooks ]
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Mixed Finite-Element (FE)
LSQ Finite-Element -
FOSLS (FE)
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Bassi-Rebay and LDG (DG,SV,SD)
Nishikawa-Roe (RD)
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Utilization of Alternative Model:
Hyperbolic model, Relaxation model, Kinetic model, etc.
[ Note: Not to solve them, but to derive a diffusion scheme. ]
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Hyperbolic Model (All Methods)
Kinetic method (FV,RD)
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Hyperbolic Model (All Methods)
Kinetic method (DG)
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Hyperbolic System Approach:
[ Nishikawa, JCP2007 ]
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Upwind Schemes (All Methods)
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Upwind Schemes (All Methods; steady)
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LSQ: Least-Squares,
WENO: Weighted-Essentially Non-Oscillatory schemes,
RBC: Residual-Based Compact schemes,
DDG: Direct Discontinuous Galerkin method,
LDG: Local Discontinuous Galerkin method.
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To be added:
CDG method
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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
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