Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

We propose an integrable discrete model of one‐dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time‐dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self‐adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.     

An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter λ intrinsically, which can be chosen properly so that the semi‐implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method.     

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

In this article, we study an explicit scheme for the solution of sine‐Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo‐spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth‐order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo‐spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo‐spectral method for the numerical simulation of the sine‐Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long‐time numerical behavior for the sine‐Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

A numerical method for pricing European options with proportional transaction costs

In the paper, we propose a numerical technique based on a finite difference scheme in space and an implicit time-stepping scheme for solving the Hamilton–Jacobi–Bellman (HJB) equation arising from the penalty formulation of the valuation of European options with proportional transaction costs. We show that the approximate solution from the numerical scheme converges to the viscosity solution of the HJB equation as the mesh sizes in space and time approach zero. We also propose an iterative scheme for solving the nonlinear algebraic system arising from the discretization and establish a convergence theory for the iterative scheme. Numerical experiments are presented to demonstrate the robustness and accuracy of the method.     

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

The numerical solution for the one‐dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high‐order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap‐frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017     

On the long‐time stability of the Crank–Nicolson scheme for the 2D Navier–Stokes equations

In this article we study the stability for all positive time of the Crank–Nicolson scheme for the two‐dimensional Navier–Stokes equations. More precisely, we consider the Crank–Nicolson time discretization together with a general spatial discretization, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable, provided a CFL‐type condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

A parareal in time procedure for the control of partial differential equationsUne technique de contrôle d'équations aux dérivées partielles en temps pararéel

We have proposed in a previous note a time discretization for partial differential evolution equation that allows for parallel implementations. This scheme is here reinterpreted as a preconditioning procedure on an algebraic setting of the time discretization. This allows for extending the parallel methodology to the problem of optimal control for partial differential equations. We report a first numerical implementation that reveals a large interest. To cite this article: Y. Maday, G. Turinici, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387–392.     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

Order optimal preconditioners for fully implicit Runge‐Kutta schemes applied to the bidomain equations

The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge–Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time‐stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge–Kutta scheme is A‐stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one‐leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly positive‐definite matrices is introduced and analyzed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq Eq 27: 1290–1312, 2011     

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     

A multilevel characteristics method for periodic convection‐dominated diffusion problems

In this article we introduce a multilevel method in space and time for the approximation of a convection‐diffusion equation. The spatial discretization is of pseudo‐spectral Fourier type, while the time discretization relies on the characteristics method. The approximate solution is obtained as the sum of two components that are advanced in time using different time‐steps. In particular, this requires the introduction of two sets of discretized characteristics curves and of two interpolation operators. We investigate the stability of the scheme and derive some error estimates. They indicate that the high‐frequency term can be integrated with a larger time‐step. Numerical experiments illustrate the gain in computing time due to the multilevel strategy. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 107–132, 2000     

Numerical simulation of a class of fractional subdiffusion equations via the alternating direction implicit method

In this article, a new numerical technique is proposed for solving the two‐dimensional time fractional subdiffusion equation with nonhomogeneous terms. After a transformation of the original problem, standard central difference approximation is used for the spatial discretization. For the time step, a new fractional alternating direction implicit (FADI) scheme based on the L₁ approximation is considered. This FADI scheme is constructed by adding a small term, so it is different from standard FADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Finally, numerical examples show the effectiveness and accuracy of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 531–547, 2016     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical analysis of the Cahn–Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.