Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms

In this article, a parameter-uniform hybrid numerical method is presented to solve a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms. Due to these discontinuities, interior layers appear in the solution of the problem considered. The hybrid numerical method uses the standard finite difference scheme in the coarse mesh region and the cubic spline difference scheme in the fine mesh region which is constructed on piecewise-uniform Shishkin mesh. Second order one sided difference approximations are used at the point of discontinuity. Error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to validate the theoretical results.     

Unconditionally stable high-order time integration for moving mesh finite difference solution of linear convection–diffusion equations

This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability of numerical schemes. Moreover, many implicit second and higher order schemes, such as the Crank–Nicolson scheme, will lose their unconditional stability. A strategy is presented for developing temporally high-order, unconditionally stable finite difference schemes for solving linear convection–diffusion equations using moving meshes. Numerical results are given to demonstrate the theoretical findings.     

Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations

In this paper, we study the convergence behavior of the local discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection–diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least $k+2$ when piecewise $P^k$ polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612–27], in which superconvergence of DG schemes for convection equations is discussed.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

Hybrid Bernstein Block–Pulse functions for solving system of fractional integro-differential equations

This paper deals with the numerical solution of system of fractional integro-differential equations. In this work, we approximate the unknown functions based on the hybrid Bernstein Block–Pulse functions, in conjunction with the collocation method. We introduce the Riemann–Liouville fractional integral operator for the hybrid Bernstein Block–Pulse functions. This operator will be approximated by the Gauss quadrature formula with respect to the Legendre weight function and then it is utilized to reduce the solution of the fractional integro-differential equations to a system of algebraic equations. This system can be easily solved by any usual numerical methods. The existence and uniqueness of the solution have been discussed. Moreover, the convergence analysis of this algorithm will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed method in comparison with some other well-known methods.     

A unified difference-spectral method for time–space fractional diffusion equations

In this paper we develop a unified difference-spectral method for stably solving time–space fractional sub-diffusion and super-diffusion equations. Based on the equivalence between Volterra integral equations and fractional ordinary differential equations with initial conditions, this proposed method is constructed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time. Numerical experiments are carried out to verify the effectiveness of the method, and demonstrate that the unified method can achieve spectral accuracy in space and second-order accuracy in time for solving two kinds of time–space fractional diffusion equations.     

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Motivated by stochastic convection–diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection–diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction and non-linearity (noise).     

An efficient parallel solution for Caputo fractional reaction–diffusion equation

The computational complexity of Caputo fractional reaction–diffusion equation is \(O(MN^2)\) compared with \(O(MN)\) of traditional reaction–diffusion equation, where \(M\) , \(N\) are the number of time steps and grid points. A efficient parallel solution for Caputo fractional reaction–diffusion equation with explicit difference method is proposed. The parallel solution, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector vector addition and constant vector multiplication. The sum of constant vector multiplication is optimized. As to the authors’ knowledge, this is the first parallel solution for Caputo fractional reaction–diffusion equation. The experimental results show that the parallel solution compares well with the analytic solution. The parallel solution on single Intel Xeon X5540 CPU runs more than three times faster than the serial solution on single X5540 CPU core, and scales quite well on a distributed memory cluster system.     

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.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

High-order compact boundary value method for the solution of unsteady convection–diffusion problems

In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection–diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite difference approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.     

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.     

On the numerical solution of differential equations of Lane–Emden type

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane–Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane–Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Padé series form, we obtain an approximate polynomial of arbitrary order for solving Lane–Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.     

A framework for exploring numerical solutions of advection–reaction–diffusion equations using a GPU-based approach

In this paper we describe a general purpose, graphics processing unit (GP-GPU)-based approach for solving partial differential equations (PDEs) within advection–reaction–diffusion models. The GP-GPU-based approach provides a platform for solving PDEs in parallel and can thus significantly reduce solution times over traditional CPU implementations. This allows for a more efficient exploration of various advection–reaction–diffusion models, as well as, the parameters that govern them. Although the GPU does impose limitations on the size and accuracy of computations, the PDEs describing the advection–reaction–diffusion models of interest to us fit comfortably within these constraints. Furthermore, the GPU technology continues to rapidly increase in speed, memory, and precision, thus applying these techniques to larger systems should be possible in the future. We chose to solve the PDEs using two numerical approaches: for the diffusion, a first-order explicit forward Euler solution and a semi-implicit second order Crank–Nicholson solution; and, for the advection and reaction, a first-order explicit solution. The goal of this work is to provide motivation and guidance to the application scientist interested in exploring the use of the GP-GPU computational framework in the course of their research. In this paper, we present a rigorous comparison of our GPU-based advection–reaction–diffusion code model with a CPU-based analog, finding that the GPU model out-performs the CPU implementation in one-to-one comparisons.