### Thursday, July 17

10:30 AM-12:30 PM

*Kresge Auditorium*

## MS60

Mimetic Finite-Difference Methods for Partial Differential Equations (Part I of II)

Numerical solutions of partial differential equations approximated with finite difference methods which preserve and mimic the fundamental properties of the equations automatically reproduce many of the integral identities, including the conservation laws, of the continuum model for the underlying physical problem. These methods can lead to a deeper understanding of how the underlying physics, used to derive the equations, can be captured by the discrete model.

We will use discrete vector analysis to derive new mimetic finite difference methods for the divergence, gradient and curl differential operators and show how these discrete operators automatically satisfy the fundamental theorems of vector and tensor analysis. We will describe how the mimetic approach can be used to construct accurate finite difference and finite element methods on nonuniform grids for solving elliptic and parabolic equations with rough coefficients.

Methods will be analyzed that preserve a discrete analog of summation by parts theorem, that are nonoscillatory for discontinuous solutions of transport equations, and methods that are able to accurately account for both diffusive and dispersive limits of conservation laws with stiff relaxation terms. We will discuss semi-discrete methods that preserve the energy of the original system and will relate the stability and accuracy of the numerical boundary conditions to the time discretization.

**Organizers: James M. Hyman and Mikhail J. Shashkov**

*Los Alamos National Laboratory*

**10:30 Mimetic Finite Difference Approximations for the Divergence, Gradient and Curl Operators**
*James M. Hyman* and Mikhail J. Shashkov, Organizers
**11:00 Discrete Analogs of Theorems in Vector Analysis**
- James M. Hyman and
*Mikhail J. Shashkov*, Organizers
**11:30 Adaptive Methods for High-Order Difference Methods**
*Pelle Olsson*, Uppsala University, Sweden; and Margot Gerritsen, Stanford University
**12:00 Boundary Conditions for High Order Difference Methods**
- Bertil Gustafsson, Uppsala University, Sweden

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*MMD, 3/31/97*