Moving Mesh Partial Differential Equations (MMPDES) Based on the Equidistribution Principle

Huang, Weizhang; Ren, Yuhe; Russell, Robert D.

doi:10.1137/0731038

Cited by 321 publications

(309 citation statements)

References 18 publications

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“…[1,3,22,33,37,39,40]). Here, we adopt the so-called moving mesh PDE approach [31][32][33] in which a time-dependent PDE is introduced to determine the motion of the mesh. Both the moving mesh PDE (MMPDE) and the underlying physical equations are solved simultaneously or alternately.…”

Section: The Moving Meshmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

148

156

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We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic PDEs for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications. The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can follow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

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Section: The Moving Meshmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

148

156

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show abstract

“…In the one-dimensional case, the concept of equidistribution [19], [5], [11], [17] has proved useful in constructing good meshes. However, in spite of recent advances [6], [7], [18], the extension of this idea to higher-dimensional domains with a non-trivial geometry remains problematic.…”

Section: Introductionmentioning

confidence: 99%

Analysis of an algorithm for generating locally optimal meshes for 𝐿₂ approximation by discontinuous piecewise polynomials

Tourigny

Baines

1997

Math. Comp.

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Abstract. This paper discusses the problem of constructing a locally optimal mesh for the best L 2 approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M. J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

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“…The scalar parameter λ k is selected in each update to maintain this normalization. Because of the equidistribution principle (19) we see that consecutive updates will keep changing Φ N until all components of the error estimateR [k] …”

Section: Grid Generation and Refinement: The Control Algorithmmentioning

confidence: 99%

“…(In a more general setting such operations would be handled by digital filters both in time and space.) Further work on moving meshes can be found in [9] and [19]. In Section 2 we characterize the optimal grid density function associated with a given monitor function, using a variation principle.…”

mentioning

confidence: 99%

Automatic grid control in adaptive BVP solvers

2010

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Grid adaptation in two-point boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function φ(x). The local mesh width ∆x j+1/2 = x j+1 − x j with 0 = x 0 < x 1 < · · · < x N = 1 is computed as ∆x j+1/2 = ǫ N /ϕ j+1/2 , where {ϕ j+1/2 } N −1 0 is a discrete approximation to the continuous density function φ(x), representing mesh width variation. The parameter ǫ N = 1/N controls accuracy via the choice of N . For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once φ(x) is determined, another control law determines N based on the prescribed tolerance tol. The paper focuses on the interaction between control system and solver, and the controller's ability to produce a nearoptimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.

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Moving Mesh Partial Differential Equations (MMPDES) Based on the Equidistribution Principle

Cited by 321 publications

References 18 publications

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Analysis of an algorithm for generating locally optimal meshes for 𝐿₂ approximation by discontinuous piecewise polynomials

Automatic grid control in adaptive BVP solvers

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