Numerical methods for heat equation with variable coefficients

In this paper, two methods are developed for linear parabolic partial differential equation with variable coefficients, which are based on rational approximation to the matrix exponential functions. These methods are L-stable, third-order accurate in space and time. In the development of these methods, second-order spatial derivatives are approximated by third-order finite-difference approximations, which give a system of ordinary differential equations whose solution satisfies a recurrence relation that leads to the development of algorithms. These algorithms are tested on heat equation with variable coefficients, subject to homogeneous and/or time-dependent boundary conditions, and no oscillations are observed in the experiments. The method is also modified for a nonlinear problem. All these methods do not require complex arithmetic, and based on partial fraction technique, which is very useful for parallel processing.     

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection–diffusion and pseudo-parabolic equations.     

Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Particle tracking for fractional diffusion with two time scales

Previous work [Y. Zhang, M.M. Meerschaert, B. Baeumer, Particle tracking for time-fractional diffusion, Phys. Rev. E 78 (2008) 036705] showed how to solve time-fractional diffusion equations by particle tracking. This paper extends the method to the case where the order of the fractional time derivative is greater than one. A subordination approach treats the fractional time derivative as a random time change of the corresponding Cauchy problem, with a first derivative in time. One novel feature of the time-fractional case of order greater than one is the appearance of clustering in the operational time subordinator, which is non-Markovian. Solutions to the time-fractional equation are probability densities of the underlying stochastic process. The process models movement of individual particles. The evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations, or Langevin equations. Monte Carlo simulation yields particle location, and the ensemble density approximates the solution to the variable coefficient time-fractional diffusion equation in one or several spatial dimensions. The particle tracking code is validated against inverse transform solutions in the simplest cases. Further applications solve model equations for fracture flow, and upscaling flow in complex heterogeneous porous media. These variable coefficient time-fractional partial differential equations in several dimensions are not amenable to solution by any alternative method, so that the grid-free particle tracking approach presented here is uniquely appropriate.     

Diagonalization based Parallel-in-Time method for a class of fourth order time dependent PDEs

In this paper, we design, analyze and implement efficient time parallel methods for a class of fourth order time-dependent partial differential equations (PDEs), namely the biharmonic heat equation, the linearized Cahn–Hilliard (CH) equation and the nonlinear CH equation. We use a diagonalization technique on all-at-once system to develop efficient iterative time parallel methods for investigating the solution behaviour of the said equations. We present the convergence analysis of Parallel-in-Time (PinT) algorithms. We verify our findings by presenting numerical results.     

Error function scheme for Burgers' equation

The present paper is devoted to the development of a new scheme to solve the one-dimensional time-dependent Burgers' equation locally on sub-domains, using similarity reductions for partial differential equations. Each sub-domain is divided into three grid points. The ordinary differential equation deduced from the similarity reduction can be integrated and is then used to approximate the flux vector in the Burgers' equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two different test cases and are compared with other numerical results.     

Hierarchical matrix arithmetic with accumulated updates

Hierarchical matrices can be used to construct efficient preconditioners for partial differential and integral equations by taking advantage of low-rank structures in triangular factorizations and inverses of the corresponding stiffness matrices. The setup phase of these preconditioners relies heavily on low-rank updates that are responsible for a large part of the algorithm’s total run-time, particularly for matrices resulting from three-dimensional problems. This article presents a new algorithm that significantly reduces the number of low-rank updates and can shorten the setup time by 50% or more.

     

Discrete artificial boundary conditions for transient scalar wave propagation in a 2D unbounded layered media

This paper presents an efficient numerical method for direct time-domain solution of the transient scalar wave propagation in a two-dimensional unbounded multi-layer soil. The unbounded domain is truncated by an artificial boundary which demands the corresponding boundary conditions. In the new approach, only the artificial boundary is discretized into one-dimensional finite elements, yielding a new time-dependent partial differential equation (PDE) for displacements with respect to only one spatial coordinate. Factorization of the PDE and introduction of the residual radiation functions, there results a linear first-order ordinary differential equation (ODE). Its stability is ensured. The time-dependent discrete artificial boundary conditions are determined by the solution of the ODE. In general, it is local in time, but it is non-local in space. Several numerical examples are given to verify the superiority of the proposed method.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.     

Space–Time Adaptive Solution of First Order PDES

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge–Kutta–Fehlberg (RKF) method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers’ equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.     

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.     

Modeling of flexible-link manipulators with prismatic joints

The axially translating flexible link in flexible manipulators with a prismatic joint can be modeled using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, we present a nondimensional form of the Euler-Bernoulli beam equation using the concept of group velocity and present conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions lead to a time-dependent frequency equation for the translating flexible beam. We present a novel method to solve this time-dependent frequency equation by using a differential form of the frequency equation. We then present a systematic modeling procedure for spatial multi-link flexible manipulators having both revolute and prismatic joints. The assumed mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. We show, using a model-based control law, that the closed-loop dynamic response of modal variables become unstable during retraction of a flexible link, compared to the stable dynamic response during extension of the link. Numerical simulation results are presented for a flexible spatial RRP configuration robot arm. We show that the numerical results compare favorably with those obtained by using a finite element-based model.     

Solving a category of two-dimensional fractional optimal control problems using discrete Legendre polynomials

In this paper, we formulate a numerical method to approximate the solution of two-dimensional optimal control problem with a fractional parabolic partial differential equation (PDE) constraint in the Caputo type. First, the optimal conditions of the optimal control problems are derived. Then, we discretize the spatial derivatives and time derivatives terms in the optimal conditions by using shifted discrete Legendre polynomials and collocations method. The main idea is simplifying the optimal conditions to a system of algebraic equations. In fact, the main privilege of this new type of discretization is that the numerical solution is directly and globally obtained by solving one efficient algebraic system rather than step-by-step process which avoids accumulation and propagation of error. Several examples are tested and numerical results show a good agreement between exact and approximate solutions.     

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I

We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere.     

A hybrid method for the solution of first order partial differential equations arising in non-linear estimation

A method for the hybrid computer solution of the Cauchy problem for a general first order partial differential equation is presented. The method is based on the well known theory of characteristics and the solution is obtained by generating the characteristic curves of the partial differential equation. The problem is thus transformed to that of integrating a system of ordinary differential equations using different initial conditions. The organization of the computation is such that the analog computer integrates the system of ordinary differential equations whereas the digital computer commands the operation, supplies initial conditions and stores the results. The method has been illustrated by simulating two partial differential equations arising in non-linear estimation.     

Meshless LBIE formulations for simply supported and clamped plates under dynamic load

Simply supported and clamped thin elastic plates under dynamic loads are analyzed. Both harmonic and impact loads are considered. Viscous damping is taken into account. The governing partial differential equation (PDE) of fourth order is decomposed into two coupled PDEs of second order for the deflection and its Laplacian. Both equations contain time-dependent variables. The Laplace transform is used to eliminate the time dependence of the variables. Unknown Laplace transforms are computed from the local boundary integral equations. The meshless approximation based on the moving least square method is employed for the implementation. Time-dependent values are obtained by the Durbin inversion technique.     

TVD, WENO and blended BDF discretizations for Asian options

In this paper, we discuss topics for a fast and accurate solution of continuous American-style Asian option problems from computational finance. These problems lead to 2D time-dependent convection-dominated partial differential equations with a free boundary. As a pre-study for accurate discretization schemes in “asset price space” and in time, we solve numerically reference problems based on the Black-Scholes equation with small volatility and with discontinuous final conditions.     

A generalized solution of an initial-boundary value problem arising in the mechanics of discrete-continuous systems

We define a generalized solution of an initial-boundary value problem for a linear system of differential equations with one ordinary differential equation and two partial differential equations (a hybrid system of differential equations). We prove that the problem is well-posed and has a unique generalized solution. An analytical formula for the solution is found. Such systems of differential equations arise in studying discrete-continuum mechanical systems.