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In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm. [ABSTRACT FROM AUTHOR]

The current paper develops and analyzes a numerical scheme for the space–time fractional stochastic nonlinear diffusion wave equations. The implicit scheme is based on the matrix transform technique for discretizing the Riesz-space fractional derivative, (3 − α) -order approximation to the Caputo-fractional derivative in time and Taylor's series method to linearize the nonlinear source term, and has been successfully applied to solve a class of nonlinear fractional diffusion wave equation. We prove that the implicit scheme is convergent with β-order in space and (3 − α) order in time, respectively. The optimum error estimates in the temporal-spatial direction and unconditional stability of the implicit scheme have been theoretically investigated. Moreover, several specific numerical experiments confirm the consistency and high efficacy of the provided algorithms, which minimizes the computational costs. [ABSTRACT FROM AUTHOR]

In this paper, two localized meshless schemes using polynomial basis functions have been presented for solving space–time problems. The proposed methods are especially attractive for solving time-dependent problems in 3D spacious domain. The results showed the simplicity and high accuracy of the proposed methods. Six numerical examples are given to illustrate the effectiveness of the proposed methods. We also presented the pros and cons of the proposed methods by comparing to the results of some recent publications using global polynomial basis functions and localized radial basis functions. [ABSTRACT FROM AUTHOR]

The current paper proposes a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. At the first stage, the time derivative has been approximated by a difference scheme with second-order accuracy. Furthermore, in the next step, a compact operator has been employed to discretize the space derivative with fourth-order accuracy. After deriving the time-discrete scheme, its stability is analysed. So, a suitable term is added to the main difference scheme. By adding this term, we could construct the main ADI scheme. In the final stage, the convergence order of the full-discrete scheme based upon the ADI formulation is proved. The convergence order of the constructed technique is O ((h x α) 4 + (h y β) 4 + τ 2). The numerical results show the efficiency of the new technique. [ABSTRACT FROM AUTHOR]

The space-time Conservation Element Solution Element (CE/SE) method is used to solve the 1D unsteady equation of thermal diffusion in a multilayer material, with application to thermal batteries. A conservative form of the equation is derived and solved using the CE/SE method. At the interface between the layers, local thermodynamics properties are discontinuous. Their spatial derivatives are calculated using central differences scheme. First, a material with two layers is simulated. The results show a very good agreement with analytical and COMSOL Multiphysicső calculations, and the Fourier Number Insensitivity property is validated, even for values beyond the 0.5 stability criterium. In a second test, a material stack representative of a thermal battery is considered. Obtained results are also very close to the COMSOL simulations. [ABSTRACT FROM AUTHOR]

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy. [ABSTRACT FROM AUTHOR]

In this paper, by applying order reduction approach, a second-order accurate box scheme is established to solve a nonlinear delayed convection-diffusion equations with Neumann boundary conditions. By the discrete energy method, it is shown that the difference scheme is uniquely solvable, and has a convergence rate of with respect to - norm in constrained and non-constrained temporal grids. Besides, for constrained temporal step, a Richardson extrapolation method (REM) used along with the box scheme, which makes final solution third-order accurate in both time and space, is developed in detail. Finally, numerical results confirm the accuracy and efficiency of our solvers. [ABSTRACT FROM AUTHOR]

This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma (VG) process. The continuous time VG process is approximated by a continuous time process with the same first four cumulants and then discretized in time and space. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised before the expiration date. Numerical computations of European and American options are presented and compared with results obtained with finite differences method and with the Black-Scholes prices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The structural default model of Lipton and Sepp [Credit value adjustment for credit default swaps via the structural default model, J. Credit Risk 5(2) (2009), pp. 123–146] is generalized for a set of banks with mutual interbank liabilities whose assets are driven by correlated Lévy processes with idiosyncratic and common components. The multi-dimensional problem is made tractable via a novel computational method, which generalizes the one-dimensional fractional partial differential equation method of Itkin [Efficient solution of backward jump-diffusion PIDEs with splitting and matrix exponentials, J. Comput. Financ. (2014), forthcoming. Available athttp://arxiv.org/abs/1304.3159] to the two- and three-dimensional cases. This method is unconditionally stable and of the second order of approximation in space and time; in addition, for many popular Lévy models it has linear complexity in each dimension. Marginal and joint survival probabilities for two and three banks with mutual liabilities are computed. The effects of mutual liabilities are discussed, and numerical examples are given to illustrate these effects. [ABSTRACT FROM PUBLISHER]

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order ofO(τ2−α+h2), where τ andhare time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM PUBLISHER]

Blow-up solutions for the generalized Davey-Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed. [ABSTRACT FROM AUTHOR]

The advection-diffusion equation has a long history as a benchmark for numerical methods. The standard Galerkin finite-element method is accurate but lacks sufficient stability, whereas the least-squares method is stable but not accurate for small values of diffusion coefficient. One possibility to overcome it is to use a stabilized finite-element method such as Galerkin least-squares and this is the standard approach taken in this paper. The space-time GLS has been constructed by using both linear and quadratic B-spline shape functions. If advection dominates over diffusion the numerical solution is difficult, especially if boundary layers are to be resolved. The design of stability parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black-Scholes equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence in space is two, both for low-order and high-order standard FD schemes. To obtain a method that gives higher accuracy, we use an extra grid in a limited space- and time-domain. This new method is called FD6G2. The FD6G2 method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy, the adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method. [ABSTRACT FROM AUTHOR]