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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. 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.
 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: $(u_{xx})_i$ [ 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.
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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