High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem

We use the generalized L1 approximation for the Caputo fractional derivative, the second-order fractional quadrature rule approximation for the integral term, and a classical Crank-Nicolson alternating direction implicit (ADI) scheme for the time discretization of a new two-dimensional (2D) fractional integro-differential equation, in combination with a space discretization by an arbitrary-order orthogonal spline collocation (OSC) method. The stability of a Crank-Nicolson ADI OSC scheme is rigourously established, and error estimate is also derived. Finally, some numerical tests are given.     

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in H^j (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.     

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h⁴ + k²). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L₂ and L_∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.     

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence phenomena have been observed numerically in the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The purpose of this article is to demonstrate theoretically the superconvergent fourth‐order accuracy in the first‐order partial derivatives of the collocation solution at the partition nodes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 285–303, 1999     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation

We formulate and analyze a novel numerical method for solving a time‐fractional Fokker–Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534–1550, 2015     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

High order convergent modified nodal bi-cubic spline collocation method for elliptic partial differential equation

A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

Operator splitting for numerical solution of the modified Burgers' equation using finite element method

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L₂ and L_∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method.     

Maximum norm error estimates of the Crank-Nicolson scheme for solving a linear moving boundary problem

The Crank-Nicolson scheme is considered for solving a linear convection-diffusion equation with moving boundaries. The original problem is transformed into an equivalent system defined on a rectangular region by a linear transformation. Using energy techniques we show that the numerical solutions of the Crank-Nicolson scheme are unconditionally stable and convergent in the maximum norm. Numerical experiments are presented to support our theoretical results.     

Discrete-time orthogonal spline collocation methods for the nonlinear Schrödinger equation with wave operator

In this paper, discrete-time orthogonal spline collocation schemes are proposed for the nonlinear Schrödinger equation with wave operator. These schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods. The conservative property, the convergence, and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments. In addition, some interesting phenomena which require further theoretical analysis are discussed numerically.     

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

An orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the H^l–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

The pointwise estimates of a conservative difference scheme for Burgers' equation

In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L^∞ -norm when the initial value is bounded in H¹-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L^∞ -norm. Finally, numerical examples are carried out to verify our theoretical results.     

Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

Split-step orthogonal spline collocation (OSC) methods are proposed for one-, two-, and three-dimensional nonlinear Schrödinger (NLS) equations with time-dependent potentials. Firstly, the NLS equation is split into two nonlinear equations, and one or more one-dimensional linear equations. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, we propose three approximations by using quadrature formulae, but the split order is not reduced. Discrete-time OSC schemes are applied for the linear subproblems. In numerical experiments, many tests are carried out to prove the reliability and efficiency of the split-step OSC (SSOSC) methods. Solitons in one, two, and three dimensions are well simulated, and conservative properties and convergence rates are demonstrated. We also apply the ways of solving the nonlinear subproblems to the split-step finite difference (SSFD) methods and the time-splitting spectral (TSSP) methods, and the approximate ways still work well. Finally, we apply the SSOSC methods to solve some problems of Bose-Einstein condensates.     

A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation

A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L² and L^∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin–Bona–Mahony) and BBMB (Benjamin–Bona–Mahony–Burgers) equations and the well-known decay estimates are demonstrated for the computed solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 695–716, 1998     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

A marker method for the solution of the damped Burgers' equation

A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Multilevel augmentation methods for solving the Burgers' equation

In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank–Nicolson–Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1665–1691, 2015     

A reduced-order extrapolated approach to solution coefficient vectors in the Crank-Nicolson finite element method for the uniform transmission line equation

This work is concerned mainly with developing and testing the reduced-order extrapolated approach to the unknown coefficient vectors in the Crank-Nicolson finite element (CNFE) solutions for the uniform transmission line equation. For this objective, the CNFE functional and matrix models and the existence, stability, and errors of the CNFE solutions of the uniform transmission line equation are first derived. Then a reduced-order extrapolated CNFE (ROECNFE) matrix model is established by means of a proper orthogonal decomposition technique, and the existence, stability, and error estimates of the ROECNFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Especially, our work shows that the basis functions and accuracy of the ROECNFE matrix model are the same as those of the CNFE matrix model. Finally, some numerical tests are illustrated to computationally experimentally confirm the validity and sharpness of the ROECNFE method.