Expression templates for partial differential equations

The implementation of numerical algorithms for solving partial differential equations costs a lot of time and the optimization of such codes is difficult. These problems can be reduced by expression templates for partial differential equations. Using this concept, one can implement numerical algorithms for PDE’s in a language which is close to the mathematical language. Expression templates give the compiler global informations on an expression such that the optimization of the code can be left to the compiler. Numerical results for the Stokes operator on general domains in 3D are presented. Received: 18 July 2000 / Accepted: 27 October 2000     

Multilevel sparse grids collocation for linear partial differential equations,with tensor product smooth basis functions

Radial basis functions have become a popular tool for approximation and solution of partial differential equations (PDEs). The recently proposed multilevel sparse interpolation with kernels (MuSIK) algorithm proposed in Georgoulis et al. (2013) shows good convergence. In this paper we use a sparse kernel basis for the solution of PDEs by collocation. We will use the form of approximation proposed and developed by Kansa (1986). We will give numerical examples using a tensor product basis with the multiquadric (MQ) and Gaussian basis functions. This paper is novel in that we consider space–time PDEs in four dimensions using an easy-to-implement algorithm, with smooth approximations. The accuracy observed numerically is as good, with respect to the number of data points used, as other methods in the literature; see Langer (2016) and Wang et al. (2016).     

High performance computing for partial differential equations

When representing realistic physical phenomena by partial differential equations (PDE), it is crucial to approximate the underlying physics correctly, to get precise results, and to efficiently use the computer architecture. Incorrect results can appear in incompressible Navier–Stokes or Stokes problems when the numerical approach couples into spurious modes. In Maxwell or magnetohydrodynamic (MHD) equations the so-called spectrum pollution effect can occur, and the numerical solution does not stably converge to the physical one. Problems coming from a mesh that is not adapted to the underlying physical problem, or from an inadequate choice of the dependent and independent variables can lead to low precision. Efficiency of a code implementation can be improved by well adapting the parallel computer to the application. A new monitoring system enables to detect poor implementations, to find best suited resources to execute the job, and to adapt the processor frequency during.     

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.     

Nonlinear difference schemes for linear partial differential equations

We discuss a nonlinear difference scheme for approximating the solution of the initial value problem for linear partial differential equations. At each time step of the calculation the method proceeds by processing the data and determining the best possible scheme to use for that step, according to an optimization criterion to be described. We show that the method is stable and convergent applicating it on the heat equation. In all cases considered the nonlinear method was more accurate than the classical methods.     

Cyclic animation using partial differential equations

This work presents an efficient and fast method for achieving cyclic animation using partial differential equations (PDEs). The boundary-value nature associated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus created from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic animation are presented here. The first of these approaches consists of attaching the set of curves to a skeletal system, which is responsible for holding the animation for cyclic motions through a set mathematical expressions. The second approach exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation. The spine is also manipulated mathematically. In the interest of illustrating both approaches, the first one has been implemented within a framework related to cyclic motions inherent to human-like characters. Spine-based animation is illustrated by modelling the undulatory movement observed in fish when swimming. The proposed method is fast and accurate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of a point to point map. Thus, the user is offered with the choice of using either of these two animation representations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application.     

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.     

Splitting methods for fourth order parabolic partial differential equations

In this paper, several splitting methods are discussed which can be used to solve fourth order parabolic partial differential equations that are given in some suitable first order system form. The methods are generalisations of splitting methods for (second order) parabolic PDE's. For all methods which are considered, stability or instability is studied for problems in 2 and in 3 or more spatial dimensions.     

A multiprocessor architecture for solving nonlinear partial differential equations

This paper presents the architecture of a special-purpose multiprocessor system, which we call the Broadcast Cube System (BCS), for solving non-linear Partial Differential Equations (PDEs). The BCS has the following important features: (a) Being based on the conceptually simple bus interconnection scheme it is easily understood. The use of homogeneous Processing Elements (PEs) which can be realized as standard VLSI chips makes the hardware less costly. (b) The interconnection network is simple and regular. The network can easily be extended to vast number of PEs by adding buses with new PEs on them and by slightly increasing the number of PEs on existing buses. The interconnection pattern is highly redundant to support fault tolerance in the event of PE failures. (c) In terms of the switching delays, the delay a message undergoes between a pair of PEs connected to a common bus is zero. The maximum delay between any pair of PEs is one unit and thus a strong localization of communicating tasks is not needed to avoid long message delays even in networks of thousands of PEs. The effectiveness of the BCS has been demonstrated by both analytical and simulation methods using heat transfer and fluid flow simulation, which requires solution of non-linear PDEs, as a benchmark program.     

USFKAD: An expert system for partial differential equations

The computation of the solution, by the separation of variables process, of the Poisson, diffusion, and wave equations in rectangular, cylindrical, or spherical coordinate systems, with Dirichlet, Neumann, or Robin boundary conditions, can be carried out in the time, Laplace, or frequency domains by a decision-tree process, using a library of eigenfunctions. We describe an expert system, USFKAD, that has been constructed for this purpose.

Program summary

Title of program:USFKADCatalogue identifier:ADYN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYN_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneOperating systems under which the program has been tested: Windows, UNIXProgramming language used:C++, LaTeXNo. of lines in distributed program, including test data, etc.: 11 699No. of bytes in distributed program, including test data, etc.: 537 744Memory required to execute with typical data: 1.3 MegabytesDistribution format: tar.gzNature of mathematical problem: Analytic solution of Poisson, diffusion, and wave equationsMethod of solution: Eigenfunction expansionsRestrictions concerning the complexity of the problem: A few rarely-occurring singular boundary conditions are unavailable, but they can be approximated by regular boundary value problems to arbitrary accuracy.Typical running time:1 secondUnusual features of the program: Solutions are obtained for Poisson, diffusion, or wave PDEs; homogeneous or nonhomogeneous equations and/or boundary conditions; rectangular, cylindrical, or spherical coordinates; time, Laplace, or frequency domains; Dirichlet, Neumann, Robin, singular, periodic, or incoming/outgoing boundary conditions. Output is suitable for pasting into LaTeX documents.     

Application of the differential equations method for identifying communities in sparse networks

In our previous work a new method of identifying communities in networks was presented. The method is based on a time evolution of the network according to a set of differential equations. It was applied to networks consisting of fully connected sub-networks. However, networks describing real systems are often sparse. Here the method is applied to sparse networks. The results are compared with those of an agglomerative hierarchical method based on modularity maximisation proposed by Newman and a divisive method proposed by Duch and Arenas based also on optimisation of the modularity. Obtained results show that the differential equation method usually works better than two remaining methods, allowing for more appropriate identification of the network structure.     

On solving elliptic stochastic partial differential equations

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.     

Automatically determining symmetries of partial differential equations

A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

A convex penalty for switching control of partial differential equations

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau–Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples.     

Two adaptive wavelet algorithms for non-linear parabolic partial differential equations

This paper is concerned with the construction of wavelet based adaptive algorithms for the numerical resolution of evolution equations. The adaptivity is applied into two complementary directions. The first direction shares the approaches involved in classical adaptive finite element methods and is related to a solution dependent definition of spaces of approximation. The second direction is related to the approximation of evolution operators that is made solution dependent following the philosophy of essentially non-oscillatory schemes. After the construction of the schemes, numerical tests are provided.     

基于偏微分方程的图像去噪综合模型

介绍了基于偏微分方程（PDE）的两种去噪模型,即ROF模型和LLT模型。根据对这两种模型的比较,提出了应用权函数来合并ROF模型和LLT模型的综合模型。实验表明,综合模型既能克服ROF模型和LLT模型的缺点,又能融合它们的优点,在去噪、保护平滑区域、保护边缘和纹理细节方面都有较好的表现。