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}
}
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… 

Figures and Tables from this paper

Analysis Of Moving Mesh Partial Differential Equations With Spatial Smoothing
TLDR
It is shown that under weak conditions the basic property of no node-crossing is preserved by the spatial smoothing, and a local quasi-uniformity property of the coordinate transformations determined by these MMPDEs is proven.
Practical aspects of formulation and solution of moving mesh partial differential equations
TLDR
Several practical aspects of formulating and solving MMPDEs are studied, including spatial balance, scaling invariance, effective control of mesh concentration, bounds on time steps, multiple sub-steps, and two-level mesh movement.
Stability of Moving Mesh Systems of Partial Differential Equations
TLDR
Failures and successes of the moving mesh method applied to three reaction-diffusion problems are explained via an analysis of the stability and accuracy of themoving mesh PDE.
Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations
TLDR
The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method.
A Moving Mesh Framework Based on Three Parameterization Layers for 1d PDEs
TLDR
This work develops a general moving mesh framework for 1d PDEs that is based on three parameterization layers representing referential, computational and desired parameters.
Moving Mesh Strategy Based upon a Heat FlowEquation for Two Dimensional
TLDR
The results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality.
Moving Mesh Methods Based on Moving Mesh Partial Differential Equations
TLDR
Several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh PDEs derived in a previous paper, which clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 44 REFERENCES
Moving Mesh Techniques Based upon Equidistribution, and Their Stability
TLDR
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.
TLDR
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
Equidistributing principles in moving finite element methods
A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations
TLDR
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
TLDR
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
...
1
2
3
4
5
...