Parallel ADI Smoothers for Multigrid

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics. In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

Multilevel methods offer the best promise to attain both fast convergence and parallel efficiency in the numerical solution of parabolic and elliptic partial differential equations. Unfortunately, they have not been widely used in part because of implementation difficulties for unstructured mesh solvers. To facilitate use, a multilevel preconditioner software module, ML, has been constructed. Several methods are provided requiring relatively modest programming effort on the part of the application developer. This report discusses the implementation of one method in the module: a two‐level Krylov–Schwarz preconditioner. To illustrate the use of these methods in computational fluid dynamics (CFD) engineering applications, we present results for 2D and 3D CFD benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.

The alternating group explicit (AGE) method [1] has been applied to a variety of problems involving parabolic, hyperbolic and elliptic partial differential equations. The method has been shown to be extremely powerful and flexible and to be very suitable for use on vector and parallel computers. The practical implementation of it is considered here. The new scheme is shown to possess the same convergence property with a reduced computational effort.

We compare the relative performances of two iterative schemes based on projection techniques for the solution of large sparse nonsymmetric systems of linear equations, encountered in the numerical solution of partial differential equations. The Block–Symmetric Successive Over‐Relaxation (Block‐SSOR) method and the Symmetric–Kaczmarz method are derived from the simplest of projection methods, that is, the Kaczmarz method. These methods are then accelerated using the conjugate gradient method, in order to improve their convergence. We study their behavior on various test problems and comment on the conditions under which one method would be better than the other. We show that while the conjugate‐gradient‐accelerated Block‐SSOR method is more amenable to implementation on vector and parallel computers, the conjugate‐gradient accelerated Symmetric–Kaczmarz method provides a viable alternative for use on a scalar machine.

Different splitting methods have been playing an important role in computations of numerical solutions of partial differential equations. Modern numerical strategies including mesh adaptations, linear and nonlinear transformations are also utilized together with splitting algorithms in applications. This survey concerns two cornerstones of the splitting methods, that is, the Alternating Direction Implicit (ADI) and Local One-Dimensional (LOD) methods, as well as their applications together with an eikonal mapping for solving highly oscillatory paraxial Helmholtz equations in slowly varying envelope approximations of active laser beams. The resulted finite difference scheme is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in optical wave applications.

A general method for computational fluid dynamics with boundaries moving in any prescribed fashion, is presented. The method adapts the boundary fitted mesh to the changing spatial domain by deforming it and, when necessary due to grid quality requirements, regenerating it. Finite volumes are used for the discretisation of Navier-Stokes equations, the mesh is regenerated by the advancing front method and an elliptic differential equation, or an analytical expression, is used to compute the mesh motion. Results are presented for two model problems. I. Introduction In many industrial application it is necessary to compute the flow field in a domain whose boundary is moving. One large class of such problem contains a rotating part, with applications to turbo-machinery and ship and air craft propellers. Another special problem type is piston motion in engines and hydraulic, pneumatic systems. There are also, of course, many problems which have moving boundaries and do not fall into the two mentioned restricted classes. We mention only the movement of rudders on a ship or an air plane and flows in blood vessels but the examples are countless. In this article we describe an implementation of a general method, for incompressible flow, which is applicable, in principle, to all of the problems mentioned above. The computational problem is formulated precisely in section II. The basis for the method is a finite volume approximation with an unstructured, boundary fitted, moving grid, as described in Demirdzic and Peric. 4 The new development is to dynamically determine where and when the mesh motion causes too large decrease in the mesh quality, and then regenerate the mesh for an appropriate region, interpolate the flow variables and continue the solution. The main components of the method are: The finite volume discretisation, the advancing front meshing algorithm, the method to deform the mesh and the interpolation scheme used at the mesh switching/regeneration. These steps are described in section III. The method is applied to two model problems for which we present results in section IV. We have performed a grid refinement study which indicates the appropriate convergence. We have also compared different methods for grid deformation. The first computes the grid motion by the solution of an elliptic partial differential equation and the second bases the node displacements on a weighting using the distance to the moving boundary. The results indicate that the elliptic operator, with a certain form of the coefficient function, gives slowest decrease in mesh quality. The coefficient is chosen to be large at the moving boundary, see section IV for the exact expression. This counter-acts the well-known tendency of the grid to deteriorate in this region. Throughout this article, we use the terms grid and mesh interchangeably. Another commonly used possibility for problems with rotating parts is to let the moving grid “slide” along the fixed surrounding grid, see e.g. Demirdzic, Muzaferija, Peric and Schreck. 3 The method presented here however has several advantages compared to this. First all cells in the grid has matching interfaces, there are no “hanging nodes”. Sliding meshes also necessitates book-keeping concerning which faces overlap at the sliding interface. Furthermore, the method presented here can be applied to problems involving more general boundary motion than rotation. The drawbacks with our method compared to sliding meshes are that we must regenerate the grid, and we need an algorithm for the mesh motion. Thus it is a trade-off which method to prefer. Finally, we mention the approach using overlapping (Chimera) grids as an alternative for the class of problems considered here, see e.g. Chesshire and Henshaw. 2

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

<div class="htmlview paragraph">Many methods have been suggested for the solution of parabolic partial differential equations. Prior efforts have heavily focused on explicit difference equations, implicit difference equations, and a combination of these two methods. Some work has been done in exponential methods and other obscure methods. This paper introduces an advanced exponential method for obtaining solutions of parabolic partial differential equations.</div> <div class="htmlview paragraph">The explicit, implicit, Crank-Nicolson, DuFort-Frankel, modified DuFort-Frankel, original exponential method, and the advanced exponential method are briefly described and compared. The advanced exponential method is unconditionally stable with whatever time step is taken. Accuracy and speed of solution for this method is compared to other solution methods for various example test cases. For the example test cases, the method gave excellent results when compared to the analytical solution and to other methods. The method has been introduced into the Systems Improved Numerical Differencing Analyzer and Fluid Integrator (SINDA/FLUINT) code as a solver. The advanced exponential method gives excellent speed, accuracy, and stability for solving parabolic partial differential equations.</div>

This paper proposes an asymptotic embedding method for the dynamic reconstruction of spatially varying fields. By assuming that the spatial field is the solution to an elliptic partial differential equation, then the elliptic PDE is embedded into a parabolic PDE which represents the time-varying estimator. An important advantage of the dynamic estimation scheme is the significant reduction in the use of sensing devices needed to reconstruct the spatial field. Static estimation schemes impose stringent conditions on the regularity of a regression matrix, which links the basis functions to the number of measurements. To further improve the performance of the dynamic estimator, a guidance scheme is proposed that repositions mobile sensors within the spatial field, which is linked to the performance of the dynamic estimator. Extensions to collaborative estimation and optimization of the placement of static sensors are also summarized to provide an integrated account on all facets of optimal dynamic estimation of spatial fields. Numerical simulations for spatial fields in one and two spatial dimensions are included along with a comparison of static reconstruction as quantified by the number of sensing devices required and the relative error.

Alternating direction implicit preconditioning methods for self adjoint elliptic differential equations

The solution of elliptic partial differential equations (PDEs) with sharp interfaces presents significant challenges, especially when the domain is irregular or complex, as gradients may not be well-defined at corners and may exhibit discontinuous derivatives. The local meshless method is a promising approach for addressing such difficulties. In [1], a Radial Basis Function collocation technique was proposed for the solution of 2nd-order elliptic interface PDEs having sharp corners, using stencils at the interface designed to handle corner discontinuities. While this method demonstrated notable improvements over conventional approaches, there are instances where accuracy falls short of expectations. In this work, we enhance the local RBF method by incorporating Pascal polynomials, aiming to improve numerical solutions for elliptic interface PDEs. We evaluate the method’s effectiveness in handling complex geometries and its adaptability to various interface shapes

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.