Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

Background Biochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities. Results In this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler τ-leap, as well as two more recent τ-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments. Conclusions The Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations.

A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems

This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.