High-Order Time-Accurate Parallel Schemes for Parabolic Singularly Perturbed Problems with Convection

The first boundary value problem for a singularly perturbed parabolic equation of convection-diffusion type on an interval is studied. For the approximation of the boundary value problem we use earlier developed finite difference schemes, ɛ-uniformly of a high order of accuracy with respect to time, based on defect correction. New in this paper is the introduction of a partitioning of the domain for these ɛ-uniform schemes. We determine the conditions under which the difference schemes, applied independently on subdomains may accelerate (ɛ-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on subdomains can in principle be used for parallelization of the computational method. Received December 3, 1999; revised April 20, 2000     

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order \(O(\tau ^{3-\alpha } + h^2), 0<\alpha <1\) are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where \(\tau \) and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.     

A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity

A high order finite element approach is applied to elastoplastic problems in two as well as in three dimensions. The element formulations are based on quadrilaterals and hexahedrals, taking advantage of the blending function method in order to accurately represent the geometry. A comparison of h- and p-extensions is drawn and it is shown that thin-walled structures commonly being analysed by dimensionally reduced elements may be consistently discretized by high order hexahedral elements leading to reliable and efficient computations even in case of physically nonlinear problems.     

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method. This research was supported by the Israel Science Foundation (grant No. 1362/04).     

Stability Analysis of Difference Methods for Parabolic Initial Value Problems

A decomposition of the numerical solution can be defined by the normal mode representation, that generalizes further the spatial eigenmode decomposition of the von Neumann analysis by taking into account the boundary conditions which are not periodic. In this paper we present some new theoretical results on normal mode analysis for a linear and parabolic initial value problem. Furthermore we suggest an algorithm for the calculation of stability regions based on the normal mode theory.     

Efficient Space-Time Spectral Methods for Second-Order Problems on Unbounded Domains

In this paper, we propose efficient space-time spectral methods for problems on unbounded domains. For this purpose, we first introduce two series of new basis functions on the half/whole line by matrix decomposition techniques. The new basis functions are mutually orthogonal in both \(L^2\) and \(H^1\) inner products, and lead to diagonal systems for second order problems with constant coefficients. Then we construct efficient space-time spectral methods based on Laguerre/Hermite-Galerkin methods in space and dual-Petrov-Galerkin formulations in time for problems defined on unbounded domains. Using these suggested methods, higher accuracy can be obtained. We also demonstrate that the use of simultaneously orthogonal basis functions in space may greatly simplify the implementation of the space-time spectral methods.     

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.     

Numerical Solution to a Class of Inverse Problems for Parabolic Equation

A class of inverse problems for parabolic equation is considered. In particular, boundary value problems with nonlocal conditions are reduced to such class of problems. The proposed numerical approach is based on the method of lines to reduce the problem to a system of ordinary differential equations. To solve this system, an analog of the transfer method for boundary conditions is applied. The results of numerical experiments are given.     

ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D

The ADER scheme for solving systems of linear, hyperbolic partial differential equations in two-dimensions is presented in this paper. It is a finite-volume scheme of high order in space and time. The scheme is explicit, fully discrete and advances the solution in one single step. Several numerical tests have been performed. In the first test case the dissipation and dispersion behaviour of the schemes are studied in one space dimension. Dispersion as well as dissipation effects strongly influence the discrete wave propagation over long distances and are very important for, e.g., aeroacoustical calculations. The next test, the so-called co-rotating vortex pair, is a demonstration of the ideas of the two-dimensional ADER approach. The linearised Euler equations are used for the simulation of the sound emitted by a co-rotating vortex pair.     

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized by small details in the computational domain or microstructures. In this setting, it is necessary to define a suitable coarse-level solver, in order to guarantee the scalability of the preconditioner under mesh refinement. To this end, we exploit recent ideas developed in the so-called composite finite element framework, which allows for the definition of finite element methods on general meshes consisting of agglomerated elements. Numerical experiments highlighting the practical performance of the proposed preconditioner are presented.     

High Order A-stable Numerical Methods for Stiff Problems

Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge–Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge–Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order     

Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems

We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.     

Selecting the Numerical Flux in Discontinuous Galerkin Methods for Diffusion Problems

In this paper we present numerical investigations of four different formulations of the discontinuous Galerkin method for diffusion problems. Our focus is to determine, through numerical experimentation, practical guidelines as to which numerical flux choice should be used when applying discontinuous Galerkin methods to such problems. We examine first an inconsistent and weakly unstable scheme analyzed in Zhang and Shu, Math. Models Meth. Appl. Sci. (M ³ AS) 13, 395–413 (2003), and then proceed to examine three consistent and stable schemes: the Bassi–Rebay scheme (J. Comput. Phys. 131, 267 (1997)), the local discontinuous Galerkin scheme (SIAM J. Numer. Anal. 35, 2440–2463 (1998)) and the Baumann–Oden scheme (Comput. Math. Appl. Mech. Eng. 175, 311–341 (1999)). For an one-dimensional model problem, we examine the stencil width, h-convergence properties, p-convergence properties, eigenspectra and system conditioning when different flux choices are applied. We also examine the ramifications of adding stabilization to these schemes. We conclude by providing the pros and cons of the different flux choices based upon our numerical experiments.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Numerical Solution of Inverse Problems of Thermoelasticity for a Composite Cylinder

The thermoelastic state of a composite cylinder is analyzed. Classical generalized problems defined on classes of discontinuous functions are presented. Explicit expressions are obtained for residual gradients (using the solution of direct and adjoint problems) for the implementation of Alifanov’s gradient methods; functions of the finite element method are used to construct highly accurate computation schemes for the numerical discretization of direct and adjoint problems.     

C 0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

C ⁰ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A post-processing procedure that can generate C ¹ approximate solutions from the C ⁰ approximate solutions is presented. New C ⁰ interior penalty methods based on the techniques involved in the post-processing procedure are introduced. These new methods are applicable to rough right-hand sides.Susanne C. Brenner: The work of S.C. Brenner was supported in part by the National Science Foundation under Grant Nos. DMS-00-74246 and DMS-03-11790. This revised version was published online in July 2005 with corrected volume and issue numbers.     

Composite Distributive Lattices as Annotation Domains for Mediators

In a mediator system based on annotated logics it is a suitable requirement to allow annotations from different lattices in one program on a per-predicate basis. These lattices however may be related through common sublattices, hence demanding predicates which are able to carry combinations of annotations, or access to components of annotations. We show both demands to be satisifiable by using various composition operations on the domain of complete bounded distributive lattices or bilattices, most importantly the free distributive product. An implementation of the presented concepts, based on the KOMET implementation of SLG-AL with constraints, is briefly introduced.