This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.
MR schemes for hyperbolic partial differential equations were first proposed by Harten @cite_0 . We refer to the work of M "uller @cite_26 for a survey on MR methods, see also @cite_40 . As stated above, the idea behind the MR method is to accelerate a reference discretization scheme while controlling the error. In the context of fully adaptive MR methods @cite_21 , the mathematical analysis is complete only in the case of a scalar conservation law, but in practice, these techniques have been used by several groups (see e.g. @cite_30 @cite_10 @cite_26 @cite_3 @cite_6 ) to successfully solve a wide class of problems, including applications to multidimensional systems. For more details on the framework of classical MR methods for hyperbolic partial differential equations, we also refer to @cite_21 and @cite_35 .
@cite_30 @cite_35 @cite_26 @cite_21 @cite_3 @cite_6 @cite_0 @cite_40 @cite_10
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We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers. Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points. In this paper the extension of the GFD to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one, two or three space dimensions. The convergence of the method has been studied and the truncation errors over irregular grids are given. Different examples have been solved using the explicit finite difference formulae and the criterion of stability. This has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one, two or three space dimensions. The numerical results show the accuracy obtained over irregular grids. This paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time-dependent equations. A new code for astrophysical magnetohydrodynamics (MHD) is described. The code has been designed to be easily extensible for use with static and adaptive mesh refinement. It combines higher order Godunov methods with the constrained transport (CT) technique to enforce the divergence-free constraint on the magnetic field. Discretization is based on cell-centered volume averages for mass, momentum, and energy, and face-centered area averages for the magnetic field. Novel features of the algorithm include (1) a consistent framework for computing the time- and edge-averaged electric fields used by CT to evolve the magnetic field from the time- and area-averaged Godunov fluxes, (2) the extension to MHD of spatial reconstruction schemes that involve a dimensionally split time advance, and (3) the extension to MHD of two different dimensionally unsplit integration methods. Implementation of the algorithm in both C and FORTRAN95 is detailed, including strategies for parallelization using domain decomposition. Results from a test suite which includes problems in one-, two-, and three-dimensions for both hydrodynamics and MHD are given, not only to demonstrate the fidelity of the algorithms, but also to enable comparisons to other methods. The source code is freely available for download on the web. We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem. *Research of the first author is suppo...