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Adaptivity with moving grids
, 2009
"... In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partia ..."
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Cited by 28 (5 self)
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In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes. In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh. We show that, guided by these, we can design effective moving mesh strategies. We then look in more detail as to how these strategies are implemented. Firstly we look at positionbased methods and the use of moving mesh partial differential equation (MMPDE), variational and optimal transport methods. This is followed by an analysis of velocitybased methods such as the geometric conservation law (GCL) methods. Finally we look at a number of examples where the use of a moving mesh method is effective in applications. These include scaleinvariant problems, blowup problems, problems with moving fronts and problems in meteorology. We conclude that, whilst radaptive methods are still in a relatively new stage of development, with many outstanding questions remaining, they have enormous potential for development, and for many problems they represent an optimal form of adaptivity.
Moving mesh finite element methods for the incompressible Navier–Stokes equations
 SIAM J. Sci. Comput
, 2005
"... Abstract. This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergencefree for the velocity field at each ti ..."
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Cited by 23 (6 self)
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Abstract. This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergencefree for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. Li, T. Tang, and P. W. Zhang, J. Comput. Phys., 170 (2001), pp. 562–588], which decouples the PDE solver and the meshmoving algorithm. This approach requires interpolating the solution on the newly generated mesh. Designing a divergencefreepreserving interpolation algorithm is the first goal of this work. Selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations, which is the second goal of this study. The performance of the moving mesh scheme is tested on the standard periodic double shear layer problem. No spurious vorticity patterns appear when even fairly coarse grids are used.
An adaptive mesh redistribution method for nonlinear HamiltonJacobi . . .
, 2002
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Mathematical principles of anisotropic mesh adaptation
 Commun. Comput. Phys
, 2006
"... Abstract. Mesh adaptation is studied from the mesh control point of view. Two principles, equidistribution and alignment, are obtained and found to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. A key component in these principles is the moni ..."
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Cited by 22 (6 self)
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Abstract. Mesh adaptation is studied from the mesh control point of view. Two principles, equidistribution and alignment, are obtained and found to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. A key component in these principles is the monitor function, a symmetric and positive definite matrix used for specifying the mesh information. A monitor function is defined based on interpolation error in a way with which an error bound is minimized on a mesh satisfying the equidistribution and alignment conditions. Algorithms for generating meshes satisfying the conditions are developed and twodimensional numerical results are presented.
Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction
 SIAM J. Numer. Anal
, 2007
"... Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of ..."
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Cited by 15 (7 self)
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Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000)] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005)]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunovtype finite volume schemes. Overlapping cells lend themselves to the development of a centraltype discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comput. Phys. 141 (1998)] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunovtype schemes. In particular, the
Moving mesh generation using the Parabolic MongeAmpère equation
, 2008
"... This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear pa ..."
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Cited by 10 (4 self)
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This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing radaptivity based on ideas from Optimal Transportation combined with the equidistribution principle applied to a (time varying) scalar monitor function (used successfully in moving mesh methods in onedimension). Detailed analysis of this new method is presented in which the convergence, regularity and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar timedependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial
Secondorder Godunovtype scheme for reactive flow calculations on moving meshes
, 2005
"... The method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered. The flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ODE part. The numerical algorithm is based on the Godunov’s scheme on defo ..."
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Cited by 9 (2 self)
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The method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered. The flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ODE part. The numerical algorithm is based on the Godunov’s scheme on deforming meshes with some modification to increase the schemeorder in time and space. The variational approach is applied to generate the moving adaptive mesh. At every time step the functional of smoothness, written on the graph of the control function, is minimized. The gridlines are condensed in the vicinity of the main solution singularities, e.g. precursor shock, fire zones, intensive transverse shocks, and slip lines, which allows resolving a fine structure of the reaction domain. The numerical examples relating to the ChapmanJouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions.
A general moving mesh framework in 3D and its application for simulating the mixture of multiphase flows
 Commun. Comput. Phys
"... Abstract. In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical syst ..."
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Cited by 8 (1 self)
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Abstract. In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical systems while minimizing computational costs. The algorithm is a generalization of the moving mesh methods based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562588, and 177 (2002), pp. 365393]. To make 3D moving mesh simulations possible, the key is to develop an efficient mesh redistribution procedure so that this part will cost as little as possible comparing with the solution evolution part. Since the mesh redistribution procedure normally requires to solve large size matrix equations, we will describe a procedure to decouple the matrix equation to a much simpler blocktridiagonal type which can be efficiently solved by a particularly designed multigrid method. To demonstrate the performance of the proposed 3D moving mesh strategy, the algorithm is implemented in finite element simulations of fluidfluid interface interactions in multiphase flows. To demonstrate the main ideas, we consider the for
Nonoscillatory hierarchical reconstruction for central and finite volume schemes
 Comm. Comput. Phys
"... Abstract. This is the continuation of the paper ”Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction ” by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite v ..."
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Cited by 7 (5 self)
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Abstract. This is the continuation of the paper ”Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction ” by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on nonstaggered grids. This takes a new finite volume approach for approximating nonsmooth solutions. A critical step for highorder finite volume schemes is to reconstruct a nonoscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially nonoscillatory solutions even for 5thorder schemes without characteristic decomposition.
FOURIER–PADÉ APPROXIMATIONS AND FILTERING FOR SPECTRAL SIMULATIONS OF AN INCOMPRESSIBLE
"... Abstract. In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the wellknown Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic pie ..."
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Cited by 7 (3 self)
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Abstract. In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the wellknown Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier–Padé–Galerkin and Fourier–Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase. Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the onedimensional inviscid Burgers ’ equation and the twodimensional incompressible inviscid Boussinesq convection flow. 1.