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ABELIAN SANDPILES AND THE HARMONIC MODEL
, 2009
"... We present a construction of an entropypreserving equivariant surjective map from the ddimensional critical sandpile model to a certain closed, shiftinvariant subgroup of T Zd (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove un ..."
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We present a construction of an entropypreserving equivariant surjective map from the ddimensional critical sandpile model to a certain closed, shiftinvariant subgroup of T Zd (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.
Windows 2000 clustering: Performing a rolling upgrade. http://www.microsoft.com/windows2000/ techinfo/planning/incremental/roll% upgr.asp
, 2000
"... Abstract. We propose an FFTbased algorithm for computing fundamental solutions of difference operators with constant coefficients. Our main contribution is to handle cases where the symbol has zeros. Key words. Discrete fundamental solutions, fast Fourier transform AMS subject classifications. Prim ..."
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Abstract. We propose an FFTbased algorithm for computing fundamental solutions of difference operators with constant coefficients. Our main contribution is to handle cases where the symbol has zeros. Key words. Discrete fundamental solutions, fast Fourier transform AMS subject classifications. Primary 65T50, 39A70; Secondary 65N22, 65F10 1. Background. Fundamental solutions play an important role in the theory of linear partial differential equations. Among other things, fundamental solutions can be used to prove existence and investigate regularity of solutions, see for example the work by Hörmander, in particular [10]. Fundamental solutions have the potential of playing an equally important role in the theory of difference equations, although this area of research has not achieved the same amount of attention. One important result however, is due to de Boor, Höllig, and Riemenschneider. In 1989, they proved that every partial difference operator with constant coefficients has a fundamental solution that grows no faster than a polynomial [6].
DISCRETE FUNDAMENTAL SOLUTION PRECONDITIONING FOR HYPERBOLIC SYSTEMS OF PDE
"... Abstract. We present a new preconditioner for the iterative solution of linear systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow a ..."
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Abstract. We present a new preconditioner for the iterative solution of linear systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electormagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple onestage iterative method. As an example of a more involved problem, we consider the steady state solution of the nonlinear Euler equations in a two dimensional, nonaxisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity. Key words. Preconditioner, fundamental solution, Euler equations
BOUNDARY SUMMATION EQUATIONS Per Sundqvist
, 2004
"... A new solution method for systems of partial difference equations is presented. It can be seen as a discrete counterpart of boundary integral equations, but with sums instead of integrals. The number of unknowns in systems of linear difference equations with constant coefficients defined on uniform ..."
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A new solution method for systems of partial difference equations is presented. It can be seen as a discrete counterpart of boundary integral equations, but with sums instead of integrals. The number of unknowns in systems of linear difference equations with constant coefficients defined on uniform ddimensional grids are reduced so that one dimension is eliminated. The reduction is obtained using fundamental solutions of difference operators, yielding a reduced system that is dense. The storage of the reduced system requires O(N) memory positions, where N is the length of the original vector of unknowns. The application of the matrix utilizes fast Fourier transform as its most complex operation, and requires hence O(N log N) arithmetic operations. Numerical experiments are performed, exploring the behavior of GMRES when applied to reduced systems originating from discretizations of partial differential equations. Model problems are chosen to include scalar equations as well as systems, with various boundary conditions, and on differently shaped domains. The new solution method performs well for an upwind discretization of an inviscid flowproblem. A proof of grid independent convergence is given for a simpler iterative method applied to a specific discretization of a first order differential equation. The numerical experiments indicate that this property carries over to many other problems in the same class. Key words: Fundamental solutions, partial differential equations, partial difference equations, boundary methods. 1
DISCRETE FUNDAMENTAL SOLUTION PRECONDITIONING FOR HYPERBOLIC SYSTEMS OF PDE
, 2003
"... We present a new preconditioner for the iterative solution of linear systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electo ..."
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We present a new preconditioner for the iterative solution of linear systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electormagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple onestage iterative method. As an example of a more involved problem, we consider the steady state solution of the nonlinear Euler equations in a two dimensional, nonaxisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.
monodiffr2007 20080114 Dedicated to Professor Qikeng Lu on
"... the occasion of his eightieth birthday Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the second kind ..."
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the occasion of his eightieth birthday Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the second kind
GREEN’S FUNCTION OF A CENTERED PARTIAL DIFFERENCE EQUATION
"... Applying a variation of Jacobi iteration we obtain the Green’s function for the centered partial difference equation ∆wwu(xw−1,yz) + ∆zzu(xw,yz−1) + f(u(xw,yz)) = 0, which is the result of applying the finite difference method to an associated nonlinear partial differential equation of the form uxx ..."
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Applying a variation of Jacobi iteration we obtain the Green’s function for the centered partial difference equation ∆wwu(xw−1,yz) + ∆zzu(xw,yz−1) + f(u(xw,yz)) = 0, which is the result of applying the finite difference method to an associated nonlinear partial differential equation of the form uxx + uyy + h(u) = 0. We show that approximations of the partial differential equation can be found by applying fixed point theory instead of the standard techniques associated with solving a system of nonlinear equations. Key words and phrases: Finite difference method, partial difference equations, Green’s function.