Difference Schemes For Parabolic Equations Research Articles

Compact and Monotone Difference Schemes for Parabolic Equations

Compact and Monotone Difference Schemes for Parabolic Equations

Stability analysis of inverse Lax–Wendroff boundary treatment of high order compact difference schemes for parabolic equations

In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge–Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm’s stability, one is based on the Godunov–Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results.

Locally One-Dimensional Difference Schemes for Parabolic Equations in Media Possessing Memory

Many processes in complex systems are nonlocal and possess long-term memory. Such problems are encountered in the theory of wave propagation in relaxing media [1, p. 86], whose equation of state is distinguished by a noninstantaneous dependence of the pressure p(t) on the density ρ(t); the value of p at a time t is determined by the value of the density ρ at all preceding times; i.e., the medium has memory. Similar problems are also encountered in mechanics of polymers and in the theory of moisture transfer in soil [2]; the same equation arises in the theory of solitary waves [3] and is also called the linearized alternative Korteweg–de Vries equation, or the linearized Benjamin–Bona–Mahony equation. One of such problems was studied in [4]. In the present paper, a locally one-dimensional scheme for parabolic equations with a nonlocal source, where the solution depends on the time t at all preceding times, is considered.

Local-one-dimensional difference schemes for parabolic equations in media with "memory"

Local-one-dimensional difference schemes for parabolic equations in media with "memory"

Influence of complex coefficients on the stability of difference scheme for parabolic equations with non-local conditions

The stability of a finite difference scheme for Schrödinger, Kuramoto–Tsuzuki and parabolic equations, subject to non-local conditions with complex coefficients, is dealt with. The stability conditions, which have to be met by complex coefficients in non-local conditions, have been determined. The main result of this study is that complex coefficients together with non-local conditions cause new effects on the stability of difference scheme. Numerical experiment has revealed additional regularities in the stability conditions.

High-order ADI finite difference schemes for parabolic equations in the combination technique with application in finance

In this article we combine high-order (HO) finite difference discretisations with alternating direction implicit (ADI) schemes for parabolic partial differential equations with mixed derivatives in a sparse grid setting. In each implicit leg of the ADI schemes, we propose a high-order-compact (HOC) discretisation, such that only tridiagonal systems have to be solved. With the help of HO spatial discretisations and ADI schemes solutions with second order accuracy in time and fourth order accuracy in space can be computed. In order to reduce the number of involved grid points we use the combination technique to construct the so called sparse grid solution. The theoretical findings are illustrated by numerical examples with European basket options.

Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation

In this paper, the parallel difference schemes for parabolic equation are studied. The general alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation are presented and proved to be unconditionally stable by the energy method. Two kinds of alternating band explicit (ABE1, ABE2) schemes, alternating band explicit–implicit (ABE–I) scheme and alternating band Crank–Nicolson (ABC–N) scheme are the special cases of the general schemes constructed here. Numerical experiments are performed to examine the accuracy and unconditional stability.

Evolutionary factorization and superfast relaxation count

Systems of linear algebraic equations with strongly rarefied matrices of enormous size appear in finite-difference solving of multidimensional elliptic equations. These systems are solved by iteration methods that converge rather slowly. For rectangular grids with variable coefficients and net steps, a considerably faster method is proposed. In the case of difference schemes for parabolic equations, a cost effective method called evolutionary factorization is developed. For elliptic equations, a relaxation count by evolutionally factorized schemes is suggested. This iteration method has the logarithmic convergence speed. A set of steps that practically optimizes the convergence of this algorithm and a procedure to regulate the steps that resembles the Richardson method are proposed. The procedure allows one to obtain an a posteriori asymptotically accurate error estimate of the iteration process. Previously such estimates for iteration processes were not known.

On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions

We consider three-layer difference schemes for a one-dimensional linear parabolic equation with nonlocal integral conditions. A three-layer scheme is written out in an equivalent form of a two-layer scheme. We analyze the dependence of the spectrum of the difference operator on the parameters occurring in the integral conditions. We derive stability conditions for the original three-layer scheme in a specially defined energy norm.

Numerical spectral analysis of a difference operator with non-local boundary conditions

The spectrum of a finite difference operator, subject to non-local Robin type boundary conditions, is dealt with. We analyse the spectral properties that relate to the stability of finite difference schemes for parabolic equations. The impact of functions and parameters, defining non-local conditions, on a spectral structure is examined and theoretical study is supported by numerical experiments. Also, for a difference scheme, applied to a parabolic equation with non-local conditions, a sufficient stability criterion, based on spectral properties of the difference operator, is discussed. Numerical evidence suggests that such a criterion is not only sufficient for stability, but necessary, too.

On the modified Crank–Nicholson difference schemes for parabolic equation with non‐smooth data arising in biomechanics

AbstractIn the present paper, we consider the mixed problem for one‐dimensional parabolic equation with non‐smooth data generated by the blood flow through glycocalyx on the endothelial cells. Stable numerical method is developed and solved by using the r‐modified Crank–Nicholson schemes. Numerical analysis is given for a constructed problem. Copyright © 2008 John Wiley & Sons, Ltd.

On the choice of initial conditions of difference schemes for parabolic equations

AbstractWe study finite difference schemes to approximate the first initial‐boundary value problem for linear second order parabolic equations and obtain some convergence rate estimates. When difference schemes are constructed for such problems, in the process of obtaining convergence rate estimates compatible with smoothness of the solution, various authors assume that the solution of the problem can be extended to the exterior of the domain of integration, preserving the Sobolev class. Our investigations show that this restriction can be removed if, instead of using the exact initial condition, we use certain approximations of the initial conditions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S1L. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ɛ-uniformly at a rate of O(N−1lnN + N0), where N and N0 define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N−1 + N0−1 ≪ ɛ. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S1L with respect to x and t, the convergence under the condition N−1 + N0−1 ≤ ɛ1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ɛ-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ɛ-uniformly at a rate of O(N−1lnN + N0).

Stability of difference schemes for parabolic equations with dynamical boundary conditions and conditions on conjugation

In this paper we investigate the stability of two-level difference schemes for parabolic equations with dynamical boundary conditions and conditions on conjugation. Energy norms that rely on spectral problems containing the eigenvalue in the boundary conditions or conditions on conjugation are introduced. Necessary and sufficient stability conditions in these norms for weighted difference schemes are established. The introducing of appropriate discrete spectral problems enable us to precise the values of the mesh steps that control stability of the difference schemes. Numerical tests are discussed.

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω × [ 0 , T ] \Omega \times [0,T] which is decomposed into an overlapping collection of cylindrical subregions of the form Ω l ∗ × [ 0 , T ] \Omega _{l}^{\ast } \times [0,T] , for l = 1 , … , p l=1, \dotsc , p . Each of the space-time domains Ω l ∗ × [ 0 , T ] \Omega _{l}^{\ast } \times [0,T] are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters h l h_{l} and τ l \tau _{l} . In particular, the different space-time grids need not match on the regions of overlap, and the time steps τ l \tau _{l} can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit θ \theta -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument. Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

An unconditionally stable parallel difference scheme for parabolic equations

An unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper, taking the parabolic equation u t = u xx as an example. The main ideas are described as follows. We decompose the domain Ω into some overlapping subdomains, take values of the last time layer as values of the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme inside each subdomain, and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. Its unconditional stability and convergence are proved. Though its truncation error is O(1), we will still prove that the estimate order is improved to O( τ+ h).

Explicit iterative difference schemes for parabolic equations

Explicit iterative difference schemes for parabolic equations

Domain decomposition for parallelization of finite difference schemes for parabolic equations in wave propagation

Tropospheric wave propagation can be described by a parabolic differential equation which may be solved using implicit finite differences. The large grid systems which can be required in such solutions make parallelization expedient. This can be accomplished by decomposing the computational domain into several subdomains over which the solution is calculated in parallel at each step in a marching procedure. A Schwarz‐type algorithm, which has become popular in computational fluid dynamics applications, is investigated for accomplishing this domain decomposition. It is found to be unconditionally unstable for the wave propagation parabolic equation when the subdomains overlap, while for nonoverlapping subdomains, its accuracy depends strongly on the method for establishing the boundary conditions on the pseudoboundaries of the subdomains.

Majorizing difference schemes for parabolic equations on unstructured grids

Majorizing difference schemes for parabolic equations on unstructured grids

Stable solvability of Pad� difference schemes for parabolic equations in H�lder spaces

We study Pade difference scheme for the approximate solution of the Cauchy problem for parabolic equations generated by the Pade fractions Rj,l, of exponential approximation. We establish an estimate of the coerciveness of the difference schemes for j=l−2,l −1, or even j=l in a smaller space than C0α(E).