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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.
Adaptive radial basis function methods for time dependent partial differential equations
 Appl. Numer. Math.,54:79–94,2005
"... Radial basis function (RBF) methods have shown the potential to be a universal grid free method for the numerical solution of partial differential equations. Both global and compactly supported basis functions may be used in the methods to achieve a higher order of accuracy. In this paper, we take a ..."
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Cited by 17 (1 self)
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Radial basis function (RBF) methods have shown the potential to be a universal grid free method for the numerical solution of partial differential equations. Both global and compactly supported basis functions may be used in the methods to achieve a higher order of accuracy. In this paper, we take advantage of the grid free property of the methods and use an adaptive algorithm to choose the location of the collocation points. The RBF methods produce results similar to the more well known and analyzed spectral methods, but while allowing greater flexibility in the choice of grid point locations. The adaptive RBF methods are most successful when the basis functions are chosen so that the PDE solution can be approximated well with a small number of the basis functions. 1
Domain decomposition approaches for mesh generation via the equidistribution principle
, 2011
"... Abstract. Moving mesh methods based on the equidistribution principle are powerful techniques for the space–time adaptive solution of evolution problems. Solving the resulting coupled system of equations, namely the original PDE and the mesh PDE, however, is challenging in parallel. We propose in th ..."
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Cited by 9 (7 self)
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Abstract. Moving mesh methods based on the equidistribution principle are powerful techniques for the space–time adaptive solution of evolution problems. Solving the resulting coupled system of equations, namely the original PDE and the mesh PDE, however, is challenging in parallel. We propose in this paper several Schwarz domain decomposition algorithms for this task. We then study in detail the convergence properties of these algorithms applied to the nonlinear mesh PDE in one spatial dimension. We prove convergence for classical transmission conditions, and optimal and optimized variants for the generation of steady equidistributing grids. A classical, parallel, Schwarz algorithm is presented and analysed for the generation of time dependent (moving) equidistributing grids. We conclude our study with numerical experiments.
A HighOrder Global Spatially Adaptive Collocation Method for 1D Parabolic PDEs
"... In this paper, we describe a highorder solver, adaptive in space and time, for the efficient numerical solution of onedimensional parabolic PDEs. Collocation at Gaussian points is employed for the spatial discretization, using a Bspline basis. A modification of the well known DAE solver, DASSL i ..."
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Cited by 7 (1 self)
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In this paper, we describe a highorder solver, adaptive in space and time, for the efficient numerical solution of onedimensional parabolic PDEs. Collocation at Gaussian points is employed for the spatial discretization, using a Bspline basis. A modification of the well known DAE solver, DASSL is used for the time integration. An a posteriori spatial error estimate is calculated at each successful time step. A new mesh selection strategy based on an equidistribution principle is presented for controlling the spatial error which is balanced with respect to the temporal error. This new mesh adaptation algorithm is shown to be robust, and particularly efficient for problems having solutions with rapid variation.
A SCHWARZ WAVEFORM MOVING MESH METHOD
, 2007
"... An rrefinement (moving mesh) method is considered for solving time dependent partial differential equations (PDEs). The resulting coupled system, consisting of the physical PDE and a moving mesh PDE, is solved by a Schwarz waveform relaxation method. In particular, the computational spacetime dom ..."
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Cited by 7 (4 self)
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An rrefinement (moving mesh) method is considered for solving time dependent partial differential equations (PDEs). The resulting coupled system, consisting of the physical PDE and a moving mesh PDE, is solved by a Schwarz waveform relaxation method. In particular, the computational spacetime domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of PDEs on each subdomain. Dirichlet boundary conditions are used to pass solution information between neighboring regions. The efficacy of this approach is demonstrated for some model problems. For problems where the solutions evolve on disparate time scales in different regions of the spatial domain, this approach demonstrates the significant savings in computational time and effort which are possible.
A moving mesh method for timedependent problems based on Schwarz waveform relaxation
 In Domain decomposition methods in science and engineering XVII, volume 60 of Lect. Notes Comput. Sci. Eng
, 2008
"... It is well accepted that the efficient solution of complex PDEs frequently requires methods which are adaptive in both space and time. Adaptive mesh methods for PDEs may be classified into one or more of the following broad categories: ..."
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Cited by 4 (2 self)
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It is well accepted that the efficient solution of complex PDEs frequently requires methods which are adaptive in both space and time. Adaptive mesh methods for PDEs may be classified into one or more of the following broad categories:
How to adaptively resolve evolutionary singularities in differential equations with symmetry
, 2009
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Regional blowup for a higherorder semilinear parabolic equation
 Euro J. Appl. Math
"... Abstract. We study the Cauchy problem in R × R+ for onedimensional 2mthorder, m> 1, semilinear parabolic PDEs of the form (Dx = ∂/∂x) ut = (−1) m+1 D 2m x u + up−1u, where p> 1, and ut = (−1) m+1 D 2m x u + eu with bounded initial data u0(x). Specifically, we are interested in those soluti ..."
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Abstract. We study the Cauchy problem in R × R+ for onedimensional 2mthorder, m> 1, semilinear parabolic PDEs of the form (Dx = ∂/∂x) ut = (−1) m+1 D 2m x u + up−1u, where p> 1, and ut = (−1) m+1 D 2m x u + eu with bounded initial data u0(x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical secondorder parabolic equations ut = uxx + up and ut = uxx + eu from combustion theory, the blowup in their higherorder counterparts is asymptotically selfsimilar. In particular, there exist exact nontrivial selfsimilar blowup solutions, u∗(x, t) = (T −t) −1/(p−1) f(y) in the case of the polynomial nonlinearity and u(x, t) = − ln(T −t)+f(y) for the exponential nonlinearity, where y = x/(T −t) 1/2m is the backward higherorder heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non–selfadjoint higherorder linear differential operators. We show that there are at least 2 ⌊ m ⌋ nontrivial selfsimilar solutions to the full PDEs. Numerical solution of the ODEs for 2 m = 2 and 3 supports this, and the time dependent solutions of the PDEs for m = 2 are then studied by using a scale invariant adaptive numerical method. It is shown that those functions f(y), which have the simplest spatial shape (e.g., a single maximum), correspond to stable selfsimilar solutions. A further countable subset of nonsimilarity blowup patterns can be constructed by linearization and matching with similarity solutions of a firstorder Hamilton–Jacobi equation.
DALHOUSIE UNIVERSITY
, 2007
"... Permission is herewith granted to Dalhousie University to circulate and to have copied for noncommercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author The author reserves other publication rights, and neither the thesis nor extensi ..."
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Permission is herewith granted to Dalhousie University to circulate and to have copied for noncommercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author’s written permission. The author attests that permission has been obtained for the use of any copyrighted material appearing in the thesis (other than brief excerpts requiring only proper acknowledgement in scholarly writing) and that all such use is clearly acknowledged.