Wavelet-Taylor Galerkin Method for the Burgers Equation

In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L_∞ errors relative to the analytical solution together with the numerical solution are reported. AMS subject classification (2000) 65M70     

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     

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014     

A large time‐step implicit moving mesh scheme for moving boundary problems

Velocity‐based moving mesh methods update the mesh at each time level by using a velocity equation with a time‐stepping scheme. A particular velocity‐based moving mesh method, based on conservation, uses explicit time‐stepping schemes with small time steps to avoid mesh tangling. However, this can prove to be impractical when long‐term behavior of the solution is of interest. Here, we present a semi‐implicit time‐stepping scheme which manipulates the structure of the velocity equation such that it resembles a variable‐coefficient heat equation. This enables the use of maximum/minimum principle which ensures that mesh tangling is avoided. It is also shown that this semi‐implicit scheme can be extended to a fully implicit time‐stepping scheme. Thus, the time‐step restriction imposed by explicit schemes is overcome without sacrificing mesh structure. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 321–338, 2014     

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     

求解一类具有Hibert核的奇异积分方程的小波方法

1　引　　言近年来,用小波方法数值求解积分方程越来越引起人们的注意.文献[1]提出的算法可将一类积分算子所对应的矩阵稀疏化,为小波方法快速求解积分方程开辟了一条新的道路这方面的研究不仅可以深入发展小波理论和应用算法,深入发展小波方法的功效,而且对边界元方法有重要的指导意义.然而研究稳健快速的数值方法,一直是这方面研究的难点问题.本文考虑带Hilbert核的奇异积分方程q(y)=12π∫2π0f(x)ctg12(x-y)dx,y∈[0,2π],(1.1)的小波数值解法;其中f(x)∈H2π,q(y)∈H2π是以2π为周期的Holder类函数;q(y)已知,f(x)待求解;(1.1)式右…     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

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)     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

The motive of the current study is to derive pointwise error estimates for the three-step Taylor Galerkin finite element method for singularly perturbed problems. Pointwise error estimates have not been derived so far for the said method in the finite element framework. Singularly perturbed problems represent a class of problems containing a very sharp boundary layer in their solution. A small parameter called singular perturbation parameter is multiplied with the highest order derivative terms. When this parameter becomes smaller and smaller, a boundary layer occurs and the solution changes very abruptly in a very small portion of the domain. Because of this sudden change in the nature of the solution, it becomes very difficult for the numerical methods to capture the solution accurately specially in the boundary layer region. In the present study finite element analysis has been carried out for such one-dimensional singularly perturbed time dependent convection-diffusion equations. Exponentially fitted splines have been used for the three-step Taylor Galerkin finite element method to converge. Pointwise error estimates have been derived for the method and it is shown that the method is conditionally convergent of first order accurate in space and third order accurate in time. Numerical results have been presented for both the linear and nonlinear problems.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

A wavelet method for solving a class of nonlinear boundary value problems

We develop an approximation scheme for a function defined on a bounded interval by combining techniques of boundary extension and Coiflet-type wavelet expansion. Such a modified wavelet approximation allows each expansion coefficient being explicitly expressed by a single-point sampling of the function, and allows boundary values and derivatives of the bounded function to be embedded in the modified wavelet basis. By incorporating this approximation scheme into the conventional Galerkin method, the interpolating property makes the solution of boundary value problems with strong nonlinearity to be very effective and accurate. As an example, we have applied the proposed method to the solution of the Bratu-type equations. Results demonstrate a much better accuracy than most methods developed so far. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the equations.     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

A wavelet-based nested iteration–inexact conjugate gradient algorithm for adaptively solving elliptic PDEs

In Cohen et al. (Math Comput 70:27–75, 2001), a new paradigm for the adaptive solution of linear elliptic partial differential equations (PDEs) was proposed, based on wavelet discretizations. Starting from a well-conditioned representation of the linear operator equation in infinite wavelet coordinates, one performs perturbed gradient iterations involving approximate matrix–vector multiplications of finite portions of the operator. In a bootstrap-type fashion, increasingly smaller tolerances guarantee convergence of the adaptive method. In addition, coarsening performed on the iterates allow one to prove asymptotically optimal complexity results when compared to the wavelet best N-term approximation. In the present paper, we study adaptive wavelet schemes for symmetric operators employing inexact conjugate gradient routines. Inspired by fast schemes on uniform grids, we incorporate coarsening and the adaptive application of the elliptic operator into a nested iteration algorithm. Our numerical results demonstrate that the runtime of the algorithm is linear in the number of unknowns and substantial savings in memory can be achieved in two and three space dimensions.