Numerical grid generation: foundations and applications
Numerical grid generation: foundations and applications
Simple adaptive grids for 1-d initial value problems
Journal of Computational Physics
Equidistribution schemes, poisson generators, and adaptive grids
Applied Mathematics and Computation
Computer Methods in Applied Mechanics and Engineering
Adaptive grid generation from harmonic maps on Reimannian manifolds
Journal of Computational Physics
A simple adaptive grid method in two dimensions
SIAM Journal on Scientific Computing
Moving mesh methods based on moving mesh partial differential equations
Journal of Computational Physics
Moving mesh partial differential equations (MMPDES) based on the equidistribution principle
SIAM Journal on Numerical Analysis
Adaptive grid radiation hydrodynamics with TITAN
13th annual international conference of the center for nonlinear studies on Nonlinear science
Structured adaptive grid generation
Applied Mathematics and Computation - Special issue on differential equations and computational simulations I
Solution adaptive direct variational grids for fluid flow calculations
Journal of Computational and Applied Mathematics
Jacobian-Weighted Elliptic Grid Generation
SIAM Journal on Scientific Computing
Moving mesh methods with upwinding schemes for time-dependent PDEs
Journal of Computational Physics
Stability of Moving Mesh Systems of Partial Differential Equations
SIAM Journal on Scientific Computing
A multgrid Newton-Krylov method for multimaterial equilibrium radiation diffusion
Journal of Computational Physics
A Multigrid Preconditioned Newton--Krylov Method
SIAM Journal on Scientific Computing
On Newton-Krylov multigrid methods for the imcompressible Navier-Stokes equations
Journal of Computational Physics
An implicit, nonlinear reduced resistive MHD solver
Journal of Computational Physics
A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations
SIAM Journal on Scientific Computing
A 2D high-ß Hall MHD implicit nonlinear solver
Journal of Computational Physics
Journal of Computational Physics
A fully implicit, nonlinear adaptive grid strategy
Journal of Computational Physics
Cost-effectiveness of fully implicit moving mesh adaptation: a practical investigation in 1D
Journal of Computational Physics
Journal of Computational Physics
The Monge-Ampère equation: Various forms and numerical solution
Journal of Computational Physics
Grid Generation Methods
Generalized Monge-Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$
SIAM Journal on Scientific Computing
Generalized Monge-Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$
SIAM Journal on Scientific Computing
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Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of this class of techniques has been the formulation of robust, reliable mesh-motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge-Kantorovich optimization (MK). In earlier publications [1,2], the advantages of this approach with regard to these points have been demonstrated for the time-independent case. In this study, we demonstrate that Monge-Kantorovich equidistribution can in fact be used effectively in a time-stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh-motion approaches, without resorting to ad hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3,4]). We explore two distinct r-refinement implementations of MK: the direct method, where the current mesh relates to an initial, unchanging mesh, and the sequential method, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior with regard to mesh distortion and robustness. The properties of the approach are illustrated with a hyperbolic PDE, the advection of a passive scalar, in 2D and 3D. Velocity flow fields with and without flow shear are considered. Three-dimensional grid, time-step, and nonlinear tolerance convergence studies are presented which demonstrate the optimality of the approach.