A diagonal splitting method for solving semidiscretized parabolic partial differential equations

In this work, a diagonal splitting idea is presented for solving linear systems of ordinary differential equations. The resulting methods are specially efficient for solving systems which have arisen from semidiscretization of parabolic partial differential equations (PDEs). Unconditional stability of methods for heat equation and advection–diffusion equation is shown in maximum norm. Generalization of the methods in higher dimensions is discussed. Some illustrative examples are presented to show efficiency of the new methods.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler–Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh.     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

一类抛物型偏微分方程的特征中心差分方法

运用特征中心差分方法来求解一类抛物型偏微分方程.通过对网格的不均匀剖分来离散方程,得到方程的特征中心差分格式.作了H1误差估计,给出了相应的定理.数值实验表明该方法对解此类问题是高效稳定的.     

A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations

We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Controllability results for discontinuous semilinear parabolic partial differential equations

In this Note, we present new null and approximate controllability results for the discontinuous semilinear heat equation. For simplicity, we only consider the distributed control case, with the support of the control functions arbitrarily small. We obtain results of the same kind with and without discontinuities provided the nonlinear term can be written as the sum of a (regular) function and a bounded multi-function.     

A family of fourth-order parallel splitting methods for parabolic partial differential equations

A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial dderivatives are approximated by fourth-order finite-difference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional head equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13 : 357–373, 1997     

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.     

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

Application of stochastic equivalence for solving parabolic partial differential equations

An approach to solving parabolic partial differential equations based on the method of stochastic characteristics is proposed. The method allows decomposition of the numerical procedure into separate unified blocks. The approximation error and the efficiency of the method are evaluated. An example is given.     

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.     

Compact ADI method for solving parabolic differential equations

A compact alternating direction implicit (ADI) method has been developed for solving two‐dimensional parabolic differential equations. In this study, the second‐order derivatives with respect to space are discretized using the high‐order compact finite differences. The Peaceman‐Rachford ADI method is then used for developing a new ADI scheme. It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable. The method can be generalized to the three‐dimensional case and an unconditionally stable compact Douglas ADI scheme is obtained. The method is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 129–142, 2002; DOI 10.1002/num.1037     

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.     

解偏微分方程的多步-小波-Galerkin方法

推广Lax-Wendroff多步方法,建立一类新的显式和隐式相结合的多步格式,并以此为基础提出了一类显隐多步-小波-Galerkin方法,可以用来求解依赖时间的偏微分方程.不同于Taylor-Galerkin方法,文中的方案在提高时间离散精度时不包含任何新的高阶导数.由于引入了隐式部分,与传统的多步方法相比该方案有更好的稳定性,适合于求解非线性偏微分方程,理论分析和数值例子都说明了方法的有效性.     

A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.