# Solving Diffusion Equations with Rough Coefficients in Rough Grids

@article{Shashkov1996SolvingDE, title={Solving Diffusion Equations with Rough Coefficients in Rough Grids}, author={Mikhail J. Shashkov and Stanly L. Steinberg}, journal={Journal of Computational Physics}, year={1996}, volume={129}, pages={383-405} }

A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, and it is superior for problems with rough coefficients and/or skewed grids. The algorithm is derived using thesupport-operators method,which constructs discrete analogs of the divergence and flux…

## 179 Citations

### The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials

- Mathematics
- 1997

A new second-order finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non-isotropic media is constructed. On problems with rough coefficients…

### Support Operators Method for the Diffusion Equation in Multiple Materials

- Mathematics
- 2012

A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is…

### Difference Methods for PDE in Several Dimensions

- Mathematics
- 2012

Finite-difference methods for partial differential equations in several dimensions are presented by first handling the basic (parabolic) diffusion equation in two dimensions by explicit and implicit…

### Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients

- GeologyJ. Comput. Phys.
- 2006

### Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients

- Computer ScienceSIAM J. Numer. Anal.
- 2017

These new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem, which allow us to use efficient algebraic solvers such as the preconditioned conjugate gradient method.

### Mimetic Finite Difference Methods for Diffusion Equations on Unstructured Triangular Grid

- Mathematics
- 2004

A finite difference algorithm for solution of stationary diffusion equation on unstructured triangular grid has been developed earlier by a support operator method. The support operator method first…

### A finite volume method for advection-diffusion problems in convection-dominated regimes

- Computer Science
- 2008

### The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods

- Computer Science
- 2001

### A discrete operator calculus for finite difference approximations

- Mathematics, Computer Science
- 2000

## References

SHOWING 1-10 OF 71 REFERENCES

### Support-operator finite-difference algorithms for general elliptic problems

- Computer Science, Mathematics
- 1995

The algorithm, which is a combination of the method of support operators and the mapping method, overcomes certain difficulties of the individual methods, producing a high-quality algorithm for solving general elliptic problems.

### Finite difference methods for modeling porous media flows

- Computer Science
- 1994

This study determines what finite-difference algorithms are best used in numerical simulation of two-dimensional single-phase saturated porous media flows when the models have a nondiagonal symmetric tensor for the mobility that has nontrivial jump discontinuities along lines that are not aligned with the coordinate axes.

### The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension

- Mathematics
- 1995

### High order finite volume approximations of differential operators on nonuniform grids

- Computer Science
- 1992

### Conservative Finite-Difference Methods on General Grids

- Mathematics, Computer Science
- 1996

The main ideas of Finite-Difference Algorithms are applied to solving problems of Systems of Linear Algebraic Equation and Systems of Nonlinear Equations.

### Accurate discretization of gradients on non-uniform curvilinear staggered grids

- Computer Science
- 1995

It is shown how to discretize the pressure gradient in the Navier{ Stokes equations on an arbitrary non-uniform curvilinear staggered grid, such that the error is zero for constant pressure gradient.…

### Comparisons of numerical solution methods for differential equations with discontinuous coefficients

- Mathematics
- 1994

### Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities

- Physics
- 1994

The accuracy of the Darcy velocity, flux, and stream function computed from lowest-order, triangle-based, control volume and mixed finite element approximations to the two-dimensional pressure…