Richardson Extrapolation Algorithm Research Articles

Solution of two-dimensional elasticity problems using a high-accuracy boundary element method

For the two-dimensional elasticity problems, we give a uniqueness and existence analysis via the single-layer potential approach leading to a system of integral equation that contains a weakly singular operator. For its numerical solutions we describe a O(h3) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O(h3) is got and the accuracy of numerical approximations can be improved to the order of O(h5) by the Richardson extrapolation algorithm (EA) once. The efficiency of the method is illustrated by two numerical examples.

On the numerical solutions of two-dimensional scattering problems for an open arc

For the scattering problems of acoustic wave for an open arc in two dimensions, we give a uniqueness and existence analysis via the single layer potential approach leading to a system of integral equations that contains a weakly singular operator. For its numerical solutions, we describe an O(h^{3}) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O(h^{3}) is got and the Richardson extrapolation algorithm (EA) is used to improve the accuracy of numerical solutions. The efficiency of the method is illustrated by a numerical example.

Analysis of a compact multi-step ADI method for linear parabolic equation

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.

A Crank-Nicolson-type compact difference method and its extrapolation for time fractional Cattaneo convection-diffusion equations with smooth solutions

A high-order Crank-Nicolson-type compact difference method is proposed for a class of time fractional Cattaneo convection-diffusion equations with smooth solutions. The convection coefficient of the equation may be spatially variable. A suitable transformation is adopted to transform the original equation into a reaction-diffusion equation, which is then discretized by a fourth-order compact difference approximation for the spatial derivative and by a second-order Crank-Nicolson-type difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability of the method and its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. A Richardson extrapolation algorithm, including its rigorous convergence analysis, is presented. This extrapolation algorithm improves the temporal accuracy of the computed solution to the third order. An application of the proposed method to the non-smooth solution which has a weak singularity at the initial time is also discussed by introducing a correction term. Numerical results demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted L2- and L∞-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order min{2−α,1+α}, where α∈(0,1) is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACTWe present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.

Error analysis of a compact finite difference method for fourth-order nonlinear elliptic boundary value problems

This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L2, H1 and L∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L2, H1 and L∞ norms.

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

A compact LOD method and its extrapolation for two-dimensional modified anomalous fractional sub-diffusion equations

A Crank–Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann–Liouville fractional derivatives of orders (1−α) and (1−β)(0<α,β<1). The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H1- and L2-norms and the weighted L∞-norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2min{α,β} in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min{α+β,4min{α,β}} if α≠β and min{1+α,4α} if α=β. A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm.

A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations

A modified Crank–Nicolson-type compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional fractional subdiffusion equations with a time Riemann–Liouville fractional derivative of order $$(1-\alpha )$$ $$(0<\alpha <1)$$ . This method improves the known compact ADI methods in the sense that it is based on the L1 approximation for the fractional derivative, the truncation errors on all time levels have the same order of $$\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})$$ and the optimal error estimate $$\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})$$ can be easily obtained in the standard $$H^{1}$$ - and $$L^{2}$$ -norms and the weighted $$L^{\infty }$$ -norm. The unique solvability, unconditional stability and convergence of the resulting scheme are rigorously proved. A Richardson extrapolation algorithm is presented to increase the temporal accuracy from the order $$2\alpha $$ to the order $$\min \{1+\alpha , 4\alpha \}$$ . Numerical results demonstrate the accuracy of the modified compact ADI method and the high efficiency of the Richardson extrapolation algorithm.

The study of a fourth-order multistep ADI method applied to nonlinear delay reaction–diffusion equations

In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L2- and H1-norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms.

Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations

In this article, a compact multistep alternating direction implicit (ADI) method is derived for solving a class of two-dimensional (2D) nonlinear viscous wave equations. This ADI method uses the combination of second-order backward differentiation formula (BDF2) solver with approximation factorization for time integration, and fourth-order Padé approximations to the second spatial derivatives for spatial discretization. It is shown by the discrete energy method that the present ADI method has good stability, and can attain convergence rate of O(Δt2+hx4+hy4) in L2- and H1-norms. Besides, the application of a three-grid Richardson extrapolation algorithm to the ADI solution can make final solution fourth-order accurate in both time and space. Numerical results are given to demonstrate the usefulness and efficiency of the resulting algorithms.

A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H1,L2 and L∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations

The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

Extrapolation algorithm of compact ADI approximation for two-dimensional parabolic equation

In this paper, we propose a compact alternating direction implicit (ADI) scheme for solving the two-dimensional parabolic equation. The maximum norm convergence is proven, and the convergence rate is second-order in time and fourth-order in space. We develop a Richardson extrapolation algorithm to increase the accuracy to sixth-order both in time and space. Numerical experiments show the effectiveness of the method.

A family of new fourth-order solvers for a nonlinear damped wave equation

In this paper, we propose a family of new three-level compact alternating direction implicit (ADI) difference schemes for solving a linear wave equation with a nonlinear damping function. By using a fourth-order accurate scheme to approximate the exact solution at the first time level, it is shown through the energy method that these difference schemes have fourth-order accuracy in space and second-order accuracy in time with respect to H1- and L∞-norms. A class of Richardson extrapolation algorithms based on three time-grid parameters are presented to obtain approximate solution of fourth-order accuracy in both time and space in L∞-norm. Numerical experiments are performed to support our theoretical results and test the accuracy and efficiency of our algorithms.

A new fourth‐order numerical algorithm for a class of three‐dimensional nonlinear evolution equations

AbstractIn this article, a new compact alternating direction implicit finite difference scheme is derived for solving a class of 3‐D nonlinear evolution equations. By the discrete energy method, it is shown that the new difference scheme has good stability and can attain second‐order accuracy in time and fourth‐order accuracy in space with respect to the discrete H1 ‐norm. A Richardson extrapolation algorithm is applied to achieve fourth‐order accuracy in temporal dimension. Numerical experiments illustrate the accuracy and efficiency of the extrapolation algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity

AbstractBased on potential theory, Steklov eigensolutions of elastic problems can be converted into eigenvalue problems of boundary integral equations (BIEs). The kernels of these BIEs are characterized by logarithmic and Hilbert singularities. In this article, the Nyström methods are presented for obtaining eigensolutions (λ(i),u(i)), which have to deal with the two kinds of singularities simultaneously. The solutions possess high accuracy orders O(h3) and an asymptotic error expansion with odd powers. Using h3 ‐Richardson extrapolation algorithms, we can greatly improve the accuracy orders to O(h5). Furthermore, a generalized Fourier series is constructed by the eigensolutions, and then solving the elasticity displacement and traction problems involves just calculating the coefficients of the series. A class of elasticity problems with boundary Γ is solved with high convergence rate O(h5). The efficiency is illustrated by a numerical example. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone’s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h3). Using h3−Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h5), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace's equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h 3) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h 3??Richardson extrapolation algorithms are used and the accuracy order is improved to O(h 5). The efficiency of the algorithms is illustrated by numerical examples.