# 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…

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## 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.

### A diffusion operator for Lagrangian meshes

- Mathematics
- 1986

A new diffusion operator for a Lagrangian mesh of irregular quadrilaterals is described. The operator diffuses cell-centered quantities and differs from previous works both in construction and in its…

### 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…

### Solution of the diffusion equation by finite elements in hydrodynamic codes

- Physics, Computer Science
- 1988