Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. $L^2$-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.     

A parallel Aitken-additive Schwarz waveform relaxation suitable for the grid

The objective of this paper is to describe a grid-efficient parallel implementation of the Aitken–Schwarz waveform relaxation method for the heat equation problem. This new parallel domain decomposition algorithm, introduced by Garbey [M. Garbey, A direct solver for the heat equation with domain decomposition in space and time, in: Springer Ulrich Langer et al. (Ed.), Domain Decomposition in Science and Engineering XVII, vol. 60, 2007, pp. 501–508], generalizes the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems. Although the standard Schwarz waveform relaxation algorithm has a linear rate of convergence and low numerical efficiency, it can be easily optimized with respect to cache memory access and it scales well on a parallel system as the number of subdomains increases. The Aitken-like acceleration method transforms the Schwarz algorithm into a direct solver for the parabolic problem when one knows a priori the eigenvectors of the trace transfer operator. A standard example is the linear three dimensional heat equation problem discretized with a seven point scheme on a regular Cartesian grid. The core idea of the method is to postprocess the sequence of interfaces generated by the additive Schwarz wave relaxation solver. The parallel implementation of the domain decomposition algorithm presented here is capable of achieving robustness and scalability in heterogeneous distributed computing environments and it is also naturally fault tolerant. All these features make such a numerical solver ideal for computational grid environments. This paper presents experimental results with a few loosely coupled parallel systems, remotely connected through the internet, located in Europe, Russia and the USA.     

Differential quadrature domain decomposition method for a class of parabolic equations

In this paper the differential quadrature method (DQM) and the domain decomposition method (DDM) are combined to form the differential quadrature domain decomposition method (DQDDM), in which the boundary reduction technique (BRM) is adopted. The DQDDM is applied to a class of parabolic equations, which have discontinuity in the coefficients of the equation, or weak discontinuity in the initial value condition. Two numerical examples belonging to this class are computed. It is found that the application of this method to the above mentioned problems is seen to lead to accurate results with relatively small computational effort.     

Preconditioners for Schwarz relaxation methods applied to differential algebraic equations

In this paper, the authors investigate the ability of Schwarz relaxation (SR) methods to deal with large systems of differential algebraic equations (DAEs) and assess their respective efficiency. Since the number of iterations required to achieve convergence of the classical SR method is strongly related to the number of subdomains and the time step size, two new preconditioning techniques are here developed. A preconditioner based on a correction using the algebraic equations is first introduced and leads to a number of iterations independent on the number of subdomains. A second preconditioner based on a correction using the Schur complement matrix makes the convergence independent on both the number of subdomains and the integration step size. Application on European electricity network is presented to outline the performance, efficiency, and robustness of the proposed preconditioning techniques for the solution of DAEs.     

Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.     

Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems

In this paper, the problem of delay dependent error estimates for waveform relaxation methods applied to Volterra type systems of functional-differential equations of neutral type including systems of differential-algebraic equations is discussed. Under a Lipschitz condition (with delay dependent right-hand side) imposed on the so-called splitting function it is shown how the error estimates depend on the character of delays and that for this reason they are better than the known error estimates for relaxation methods. It is proved that under some assumptions the exact solution can be obtained after a finite number of steps of the iterative process, i.e., we prove that the waveform relaxation methods have the same property as the classical method of steps for solving delay-differential equations with nonvanishing delays. We also show the convergence of the waveform relaxation method without assuming that the spectral radius of the corresponding matrix related to the Lipschitz coefficients for the neutral argument is less than one.     

Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

In this paper, the continuous and discrete optimal transmission conditions for the Schwarz algorithm without overlap for the Helmholtz equation are studied. Since such transmission conditions lead to non-local operators, they are approximated through two different approaches. The first approach, called optimized, consists of an approximation of the optimal continuous transmission conditions with partial differential operators, which are then optimized for efficiency. The second approach, called approximated, is based on pure algebraic operations performed on the optimal discrete transmission conditions. After demonstrating the optimal convergence properties of the Schwarz algorithm new numerical investigations are performed on a wide range of unstructured meshes and arbitrary mesh partitioning with cross points. Numerical results illustrate for the first time the effectiveness, robustness and comparative performance of the optimized and approximated Schwarz methods on a model problem and on industrial problems.     

Moving mesh methods for solving parabolic partial differential equations

A new adaptive method is described for solving nonlinear parabolic partial differential equations with moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is selected based upon the given initial data. The mesh movement at the boundary is governed by a second monitor function, which may or may not be the same as that used to drive the interior mesh movement. The method is described in detail and a selection of computational examples are presented using different monitor functions applied to the porous medium equation (PME) in one and two space dimensions.     

Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition

Applications of the multidomain Local Fourier Basis method [1], for the solution of PDEs on parallel computers are described. The present approach utilizes, in an explicit way, the rapid (exponential) decay of the fundamental solutions of elliptic operators resulting from semi-implicit discretizations of parabolic time-dependent problems. As a result, the global matching relations for the elemental solutions are decoupled into local interactions between pairs of solutions in neighboring domains. Such interactions require only local communications between processors with short communication links. Thus the present algorithm overcomes the global coupling, inherent both in the use of the spectral Fourier method and implicit time discretization scheme.This research is supported partly by a grant from the French-Israeli Binational Foundation for 1991–1992.     

A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations

A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2^?jp ) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechies are used in the Galerkin scheme. The proposed scheme is tested with both parabolic and hyperbolic partial differential equations. The numerical results indicate the versatility and effectiveness of the proposed scheme.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations

Poor performances can be obtained from classical domain decomposition algorithms to solve advection-diffusion equations in the case of convection dominated flows. Therefore, adaptive domain decomposition have been developed for such flows. We investigate the properties of some algorithms of this kind in the framework of a finite volume/finite element discretization.This research was carried out while the author was visiting the Group of Applied Mathematics and Simulation of CRS4, and was supported by an HCM fellowship.     

Polynomial collocation using a domain decomposition solution to parabolic PDE's via the penalty method and explicit/implicit time marching

A domain decomposition method is examined to solve a time-dependent parabolic equation. The method employs an orthogonal polynomial collocation technique on multiple subdomains. The subdomain interfaces are approximated with the aid of a penalty method. The time discretization is implemented in an explicit/implicit finite difference method. The subdomain interface is approximated using an explicit Dufort-Frankel method, while the interior of each subdomain is approximated using an implicit backwards Euler's method. The principal advantage to the method is the direct implementation on a distributed computing system with a minimum of interprocessor communication. Theoretical results are given for Legendre polynomials, while computational results are given for Chebyshev polynomials. Results are given for both a single processor computer and a distributed computing system.     

Particle-field decomposition and domain decomposition in parallel particle-in-cell beam dynamics simulation

Particle-in-cell (PIC) simulation is widely used in many branches of physics and engineering. In this paper, we give an analysis of the particle-field decomposition method and the domain decomposition method in parallel particle-in-cell beam dynamics simulation. The parallel performance of the two decomposition methods was studied on the Cray XT4 and the IBM Blue Gene/P Computers. The domain decomposition method shows better scalability but is slower than the particle-field decomposition in most cases (up to a few thousand processors) for macroparticle dominant applications. The particle-field decomposition method also shows less memory usage than the domain decomposition method due to its use of perfect static load balance. For applications with a smaller ratio of macroparticles to grid points, the domain decomposition method exhibits better scalability and faster speed. Application of the particle-field decomposition scheme to high-resolution macroparticle-dominant parallel beam dynamics simulation for a future light source linear accelerator is presented as an example.     

A finite difference non-matching domain decomposition algorithm for the parabolic equation

A finite difference domain decomposition algorithm on a non-overlapping non-matching grid for the parabolic equation is discussed. The basic procedure is to define the explicit scheme at the interface points with a larger mesh spacing H, then the implicit schemes with different mesh spacings are applied on the non-matching subdomains, respectively. The stability bound is released both for the one-dimensional and two-dimensional parabolic problem. Finally, numerical experiments are also presented.     

Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

Optimized Schwarz methods without overlap for highly heterogeneous media

In this paper an original variant of the Schwarz domain decomposition method is introduced for heterogeneous media. This method uses new optimized interface conditions specially designed to take into account the heterogeneity between the sub-domains on each sides of the interfaces. Numerical experiments illustrate the dependency of the proposed method with respect to several parameters, and confirm the robustness and efficiency of this method based on such optimized interface conditions. Several mesh partitions taking into account multiple cross points are considered in these experiments.     

On correction equations and domain decomposition for computing invariant subspaces

By considering the eigenvalue problem as a system of nonlinear equations, it is possible to develop a number of solution schemes which are related to the Newton iteration. For example, to compute eigenvalues and eigenvectors of an n × n matrix A, the Davidson and the Jacobi-Davidson techniques, construct ‘good’ basis vectors by approximately solving a “correction equation” which provides a correction to be added to the current approximation of the sought eigenvector. That equation is a linear system with the residual r of the approximated eigenvector as right-hand side.One of the goals of this paper is to extend this general technique to the “block” situation, i.e., the case where a set of p approximate eigenpairs is available, in which case the residual r becomes an n × p matrix. As will be seen, solving the correction equation in block form requires solving a Sylvester system of equations. The paper will define two algorithms based on this approach. For symmetric real matrices, the first algorithm converges quadratically and the second cubically. A second goal of the paper is to consider the class of substructuring methods such as the component mode synthesis (CMS) and the automatic multi-level substructuring (AMLS) methods, and to view them from the angle of the block correction equation. In particular this viewpoint allows us to define an iterative version of well-known one-level substructuring algorithms (CMS or one-level AMLS). Experiments are reported to illustrate the convergence behavior of these methods.