Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

A second-order ADI scheme for three-dimensional parabolic differential equations

A second-order unconditionally stable ADI scheme has been developed for solving three-dimensional parabolic equations. This scheme reduces three-dimensional problems to a succession of one-dimensional problems. Further, the scheme is suitable for simulating fast transient phenomena. Numerical examples show that the scheme gives an accurate solution for the parabolic equation and converges rapidly to the steady state solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:159–168, 1998     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

Grid approximation of singularly perturbed parabolic equations in the presence of weak and strong transient layers induced by a discontinuous right-hand side

The initial value problem on a line for singularly perturbed parabolic equations with convective terms is investigated. The first-and the second-order space derivatives are multiplied by the parameters ?₁ and ?₂, respectively, which may take arbitrarily small values. The right-hand side of the equations has a discontinuity of the first kind on the set $\bar \gamma $ = [x = 0] × [0, T]. Depending on the relation between the parameters, the appearing transient layers can be parabolic or regular, and the “intensity” of the layer (the maximum of the singular component) on the left and on the right of $\bar \gamma $ can be substantially different. If the parameter ?₂ at the convective term is finite, the transient layer is weak. For the initial value problems under consideration, the condensing grid method is used to construct finite difference schemes whose solutions converge (in the discrete maximum norm) to the exact solution uniformly with respect to ?₁ and ?₂ (when ?₂ is finite and, therefore, the transient layers are weak, no condensing grids are required).     

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 MODIFIED UPWIND DIFFERENCE SCHEME FOR NONLINEAR PARABOLIC EQUATIONS IN A VARIABLE DOMAIN

The nonlinear parabolic equations in a variable domain are considered. A modified up-wind difference scheme is given in the variable domain. Stability in l norm and error estimate in norm are obtained.     

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

The discrete mollification method is a convolution‐based filtering procedure suitable for the regularization of ill‐posed problems and for the stabilization of explicit schemes for the solution of PDEs. This method is applied to the discretization of the diffusive terms of a known first‐order monotone finite difference scheme [Evje and Karlsen, SIAM J Numer Anal 37 (2000) 1838–1860] for initial value problems of strongly degenerate parabolic equations in one space dimension. It is proved that the mollified scheme is monotone and converges to the unique entropy solution of the initial value problem, under a CFL stability condition which permits to use time steps that are larger than with the unmollified (basic) scheme. Several numerical experiments illustrate the performance and gains in CPU time for the mollified scheme. Applications to initial‐boundary value problems are included. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 38–62, 2012     

A modified Landweber iteration for general sideways parabolic equations

In this paper,we introduce a modified Landweber iteration to solve the sideways parabolic equation,which is an inverse heat conduction problem(IHCP) in the quarter plane and is severely ill-posed.We shall show that our method is of optimal order under both a priori and a posteriori stopping rule.Furthermore,if we use the discrepancy principle we can avoid the selection of the a priori bound.Numerical examples show that the computation effect is satisfactory.     

On the maximum principle and its application to diffusion equations

In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi‐discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h²) and O(h⁴) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h²) uniformly convergent discrete scheme or by O(h⁴) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

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 three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach

A number of new layer methods for solving the Dirichlet problemfor semilinear parabolic equations are constructed by usingprobabilistic representations of their solutions. The methodsexploit the ideas of weak sense numerical integration of stochasticdifferential equations in a bounded domain. Despite their probabilisticnature these methods are nevertheless deterministic. Some convergencetheorems are proved. Numerical tests are presented.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

Efficient space-time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space

Considering the space-time adaptive method for parabolic evolution equations we introduced in Stevenson et al., this work discusses an implementation of the method in which every step is of linear complexity. Exploiting the tensor-product structure of the space-time cylinder, the method allows for a family of trial spaces given as spans of wavelets-in-time tensorized with finite element spaces-in-space. On spaces whose bases are indexed by double-trees, we derive an algorithm that applies the resulting bilinear forms in linear complexity. We provide extensive numerical experiments to demonstrate the linear runtime of the resulting adaptive loop.     

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.     

The smoothing property for a class of doubly nonlinear parabolic equations

We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic -Laplace equation. The result is obtained via regularization and a comparison theorem.

     

A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kru kov and prove convergence of the scheme to the unique entropy solution of the problem, up to satisfaction of one of the boundary conditions.