Practical finite‐analytic method for solving differential equations by compact numerical schemes

Methodology for development of compact numerical schemes by the practical finite‐analytic method (PFAM) is presented for spatial and/or temporal solution of differential equations. The advantage and accuracy of this approach over the conventional numerical methods are demonstrated. In contrast to the tedious discretization schemes resulting from the original finite‐analytic solution methods, such as based on the separation of variables and Laplace transformation, the practical finite‐analytical method is proven to yield simple and convenient discretization schemes. This is accomplished by a special universal determinant construction procedure using the general multi‐variate power series solutions obtained directly from differential equations. This method allows for direct incorporation of the boundary conditions into the numerical discretization scheme in a consistent manner without requiring the use of artificial fixing methods and fictitious points, and yields effective numerical schemes which are operationally similar to the finite‐difference schemes. Consequently, the methods developed for numerical solution of the algebraic equations resulting from the finite‐difference schemes can be readily facilitated. Several applications are presented demonstrating the effect of the computational molecule, grid spacing, and boundary condition treatment on the numerical accuracy. The quality of the numerical solutions generated by the PFAM is shown to approach to the exact analytical solution at optimum grid spacing. It is concluded that the PFAM offers great potential for development of robust numerical schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite‐difference schemes for transport‐dominated equations using special collocation nodes

We introduce finite‐difference schemes based on a special upwind‐type collocation grid, in order to obtain approximations of the solution of linear transport‐dominated advection‐diffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Splitting methods for Hamilton‐Jacobi equations

We explain how the exploitation of several kinds of operator splitting methods, both local and global in time, lead to simple numerical schemes approximating the solution of nonlinear Hamilton‐Jacobi equations. We review the existing local methods which have been used since the early 80's and we introduce a new method which is global in time. We show some numerical experiments. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

High‐resolution monotone schemes based on quasi‐characteristics technique

In this article, we consider a new technique that allows us to overcome the well‐known restriction of Godunov's theorem. According to Godunov's theorem, a second‐order explicit monotone scheme does not exist. The techniques in the construction of high‐resolution schemes with monotone properties near the discontinuities of the solution lie in choosing of one of two high‐resolution numerical solutions computed on different stencils. The criterion for choosing the final solution is proposed. Results of numerical tests that compare with the exact solution and with the numerical solution obtained by the first‐order monotone scheme are presented. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 262–276, 2001     

A predictor‐‐corrector scheme for the sine‐Gordon equation

A predictor–corrector scheme is developed for the numerical solution of the sine‐Gordon equation using the method of lines approach. The solution of the approximating differential system satisfies a recurrence relation, which involves the cosine function. Pade' approximants are used to replace the cosine function in the recurrence relation. The resulting schemes are analyzed for order, stability, and convergence. Numerical results demonstrate the efficiency and accuracy of the predictor–corrector scheme over some well‐known existing methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 133–146, 2000     

Indecomposable quasi‐characteristics scheme on pyramidal stencil and its application for numerical simulation of two‐phase flows through heterogeneous porous medium

A new high‐resolution indecomposable quasi‐characteristics scheme with monotone properties based on pyramidal stencil is considered. This scheme is based on consideration of two high‐resolution numerical schemes approximated governing equations on the pyramidal stencil with different kinds of dispersion terms approximation. Two numerical solutions obtained by these schemes are analyzed, and the final solution is chosen according to the special criterion to provide the monotone properties in regions where discontinuities of solutions could arise. This technique allows to construct the high‐order monotone solutions and keeps both the monotone properties and the high‐order approximation in regions with discontinuities of solutions. The selection criterion has a local character suitable for parallel computation. Application of the proposed technique to the solution of the time‐dependent 2D two‐phase flows through the porous media with the essentially heterogeneous properties is considered, and some numerical results are presented. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 44–55, 2002     

An efficient splitting technique for two‐layer shallow‐water model

We consider the numerical approximation of the weak solutions of the two‐layer shallow‐water equations. The model under consideration is made of two usual one‐layer shallow‐water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one‐layer shallow‐water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow‐water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non‐negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1396–1423, 2015     

Difference methods for computing the Ginzburg‐Landau equation in two dimensions

In this article, three difference schemes of the Ginzburg‐Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time‐splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011     

Time accurate fast wavelet‐Taylor Galerkin method for partial differential equations

We introduce the concept of fast wavelet‐Taylor Galerkin methods for the numerical solution of partial differential equations. In wavelet‐Taylor Galerkin method discretization in time is performed before the wavelet based spatial approximation by introducing accurate generalizations of the standard Euler, θ and leap‐frog time‐stepping scheme with the help of Taylor series expansions in the time step. We will present two different time‐accurate wavelet schemes to solve the PDEs. First, numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for in localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions, and nonlinear equation with a shock in solution in two dimensions. Second, our schemes deal with more regular class of problems where wavelets are not efficient procedure for data compression but we can use the good approximation properties of wavelet. Here time‐accurate schemes lead to consistent mass matrix in an explicit time stepping, which can be solved by approximate factorization techniques. Numerical experiment deals with more regular class of problems like heat equation as well as coupled linear system in two dimensions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Efficient local structure‐preserving schemes for the RLW‐type equation

The multisymplectic schemes have been used in numerical simulations for the RLW‐type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure‐preserving schemes for the RLW‐type equation, which preserve the local energy exactly on any time‐space region and can produce richer information of the original problem. The schemes will be mass‐ and energy‐preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant‐preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678–1691, 2017     

From the fitting techniques to accurate schemes for the Liouville‐Bratu‐Gelfand problem

We are interested in numerical methods for the Liouville‐Bratu‐Gelfand problem. The ideas and techniques developed here to construct the schemes are inspired from the fitted method and the so‐called compact exponentially fitted method. Some of those schemes can be viewed as extensions of both the Buckmire scheme and the standard scheme which results from the use of the standard finite‐difference procedures. We study and compare computationally the accuracy of methods introduced here. It is also mentioned that the Buckmire's techniques and the standard scheme are a particular case of the fitted method. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017     

Finite analytic numerical method for solving two‐dimensional quasi‐Laplace equation

A typical power series analytic solution of quasi‐Laplace equation in the infinitesimal angle domain around the singular point of the square cells is provided in this article. Toward the singular point, the gradient of the potential variable will tend to infinity, which is described by the first term of the power series solution. Based on this analytic solution, three finite analytic numerical methods are proposed. These methods are analogous and are constructed, respectively, when considering different numbers of the terms or using different schemes to determine the relevant parameters in the power series. Numerical examples show that all of the three finite analytic numerical methods proposed can provide rather accurate solutions than the traditional numerical methods. In contrast, when using the traditional numerical schemes to solve the quasi‐Laplace equation in a strong heterogeneous medium, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. In practical applications, subdividing each origin cell into 2 × 2 or 3 × 3 subcells is enough for the finite analytical numerical methods to get relatively accurate results. The finite analytical numerical methods are also convenient to construct the flux field with high accuracy.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1755–1769, 2014     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A compact adaptive approach for degenerate singular reaction‐diffusion equations

This article concerns a compact adaptive method for the numerical solution of nonlinear degenerate singular reaction‐diffusion equations. The partial differential equation problems exhibit strong quenching blow‐up type singularities, and are critical in numerous applications ranging from optimized internal combustion designs to oil pipeline decay predictions. Adaptive schemes of fourth order in space and second order in time are acquired and discussed. Nonuniform spatial and temporal grids are utilized through suitable adaptations. Rigorous analysis is given for the numerical stability, and computational experiments are performed to illustrate our conclusions.     

SUPG and grad‐div stabilized finite element methods for steady weakly compressible viscous flow

The extensive application of mathematical and computational methods has become an efficient and powerful approach to the investigation and solution of many problems and processes in fluid dynamics from qualitative as well as quantitative point of view. In this work a new class of advanced numerical approximation schemes to isothermal compressible viscous flow is presented. The schemes are based on an iteration between an Oseen like problem for the velocity and a hyperbolic transport equation for the density. Such schemes seem attractive for computations because they offer a reduction to simpler problems for which highly refined numerical methods either are known or can be built from existing approximation schemes to similar equations, and because of the guidance that can be drawn from an existence theory based on them. For the generalized Oseen subproblem a Taylor–Hood finite element method is proposed that is stabilized by a reduced SUPG and grad‐div technique (cf. [1, 4]) in the convection‐dominated case. Results of theoretical investigations and numerical studies are presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)