# Moving mesh partial differential equations (MMPDES) based on the equidistribution principle

@article{Huang1994MovingMP,
title={Moving mesh partial differential equations (MMPDES) based on the equidistribution principle},
author={Weizhang Huang and Yuhe Ren and Robert D. Russell},
journal={SIAM Journal on Numerical Analysis},
year={1994},
volume={31},
pages={709-730}
}
• Published 1 June 1994
• Computer Science
• SIAM Journal on Numerical Analysis
This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh equations that have been used by others. Their stability is analyzed and it is seen that a key term for most of these moving mesh PDEs is a source-like term that measures the level of equidistribution. It is shown that under weak assumptions mesh crossing cannot occur for most of them. Finally, numerical…
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## References

SHOWING 1-10 OF 44 REFERENCES
Moving Mesh Techniques Based upon Equidistribution, and Their Stability
• Mathematics, Computer Science
SIAM J. Sci. Comput.
• 1992
Various aspects of the moving mesh problem are investigated for the solution of partial differential equations (PDEs) in one space dimension, and it is shown that equidistribution implicitly corresponds to finding a solution to a PDE involving a new set of computational coordinates.
Adaptive Finite Element Methods for Parabolic Partial Differential Equations.
• Computer Science
• 1983
A finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time that is able to calculate accurate solutions with fewer elements than would be necessary with a uniform mesh.
On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems
A general theory is developed for calculating equidistributing meshes $\{ t_i \}$ for difference methods for boundary-value problems of the form $u' = f(u,t),\qquad b(u(0),u(1)) = 0.$ It is
A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations
• Computer Science
• 1986
A moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension using p-hierarchic finite elements for the temporal integration of the solution, the error estimate, and the mesh motion.
A Simple Adaptive Grid Method in Two Dimensions
• Computer Science
SIAM J. Sci. Comput.
• 1994
It is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and, based on this recognition, a local equidisting principle is developed and a discrete formulation is described for grid generation in two space dimensions.
LSODE and LSODI, two new initial value ordinary differential equation solvers
Two new packages are available for the numerical solution of the initial value problem for stiff and nonstiff systems of ordinary differential equations (ODE's). LSODE solves explicitly given ODE