A large time‐step implicit moving mesh scheme for moving boundary problems

Velocity‐based moving mesh methods update the mesh at each time level by using a velocity equation with a time‐stepping scheme. A particular velocity‐based moving mesh method, based on conservation, uses explicit time‐stepping schemes with small time steps to avoid mesh tangling. However, this can prove to be impractical when long‐term behavior of the solution is of interest. Here, we present a semi‐implicit time‐stepping scheme which manipulates the structure of the velocity equation such that it resembles a variable‐coefficient heat equation. This enables the use of maximum/minimum principle which ensures that mesh tangling is avoided. It is also shown that this semi‐implicit scheme can be extended to a fully implicit time‐stepping scheme. Thus, the time‐step restriction imposed by explicit schemes is overcome without sacrificing mesh structure. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 321–338, 2014     

Understanding Saul'yev‐type unconditionally stable schemes from exponential splitting

Saul'yev‐type asymmetric schemes have been widely used in solving diffusion and advection equations. In this work, we show that Saul'yev‐type schemes can be derived from the exponential splitting of the semidiscretized equation which fundamentally explains their unconditional stability. Furthermore, we show that optimal schemes are obtained by forcing each scheme's amplification factor to match that of the exact amplification factor. A new second‐order explicit scheme is found for solving the advection equation with the identical amplification factor as the implicit Crank–Nicolson algorithm. Other new schemes for solving the advection–diffusion equation are also derived.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1961–1983, 2014     

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.     

Robust spectral method for numerical valuation of european options under Merton's jump‐diffusion model

We propose a novel numerical method based on rational spectral collocation and Clenshaw–Curtis quadrature methods together with the “” transformation for pricing European vanilla and butterfly spread options under Merton's jump‐diffusion model. Under certain assumptions, such model leads to a partial integro‐differential equation (PIDE). The differential and integral parts of the PIDE are approximated by the rational spectral collocation and the Clenshaw–Curtis quadrature methods, respectively. The application of spectral collocation method to the PIDE leads to a system of ordinary differential equations, which is solved using the implicit–explicit predictor–corrector (IMEX‐PC) schemes in which the diffusion term is integrated implicitly, whereas the convolution integral, reaction, advection terms are integrated explicitly. Numerical experiments illustrate that our approach is highly accurate and efficient for pricing financial options.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1169–1188, 2014     

A numerical method for the Landau–Lifshitz equation with magnetostriction

We present a numerical scheme for Landau–Lifshitz–Gilbert equation coupled with the equation of elastodynamics. The considered physical model describes the behaviour of ferromagnetic materials when magnetomechanical coupling is taken into account. The time‐discretization is based on the backward Euler method with projection. In the numerical approximation, the two equations are decoupled by a suitable linearization in order to solve the magnetic and mechanic part separately. The resulting semi‐implicit scheme is linear and allows larger time‐steps than explicit methods. We prove stability and error estimates for the presented time discretization in 2D. Finally, we test the accuracy of the scheme on an academic numerical example with known exact solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Variational formulations for Vlasov–Poisson–Fokker–Planck systems

Time‐discrete variational schemes are introduced for both the Vlasov–Poisson–Fokker–Planck (VPFP) system and a natural regularization of the VPFP system. The time step in these variational schemes is governed by a certain Kantorovich functional (or scaled Wasserstein metric). The discrete variational schemes may be regarded as discretized versions of a gradient flow, or steepest descent, of the underlying free energy functionals for these systems. For the regularized VPFP system, convergence of the variational scheme is rigorously established. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

We present a parallel matrix‐free implicit finite volume scheme for the solution of unsteady three‐dimensional advection‐diffusion‐reaction equations with smooth and Dirac‐Delta source terms. The scheme is formally second order in space and a Newton–Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix‐vector product required is hardcoded without any approximations, obtaining a matrix‐free method that needs little storage and is well‐suited for parallel implementation. We describe the matrix‐free implementation of the method in detail and give numerical evidence of its second‐order convergence in the presence of smooth source terms. For nonsmooth source terms, the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long‐time simulation of calcium flow in heart cells and show its parallel scaling. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 143–167, 2015     

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion problems

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion equations are presented and analyzed. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. The linear part of the equation is discretized implicitly and the nonlinear part of the equation explicitly. The schemes are stable and very efficient. We derive optimal order error estimates. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:93–104, 2001     

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     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

Continuous time mean‐variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach

We present efficient partial differential equation methods for continuous time mean‐variance portfolio allocation problems when the underlying risky asset follows a jump‐diffusion. The standard formulation of mean‐variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one‐dimensional (1D) nonlinear Hamilton–Jacobi–Bellman (HJB) partial integrodifferential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. To preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2D impulse control problem, 1D for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi‐Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs, and constraints on the portfolio, such as maximum limits on borrowing and solvency. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 664–698, 2014     

On an incompressible model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global‐in‐time existence for the associated initial‐boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

A linear,decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

In this paper, a linear decoupled fractional time stepping method is proposed and developed for the nonlinear fluid–fluid interaction governed by the two Navier–Stokes equations. Partitioned time stepping method is applied to two‐physics problems with stiffness of the coupling terms being treated explicitly and is also unconditionally stable. As for each fluid, the velocity and pressure are respectively determined by just solving one vector‐valued quasi‐elliptic equation and the Possion equation with homogeneous Neumann boundary condition per time step. Therefore, the cost of the fluid–fluid interaction is dominant to solve four simple linear equations, which greatly reduces the computational cost of the whole system. The method exploits properties of the fluid–fluid system to establish its stability and convergence with the same results as the standard scheme. Finally, numerical experiments are presented to show the performance of the proposed method.     

New epitaxial thin‐film models and numerical approximation

This paper concerns new continuum phenomenological model for epitaxial thin‐film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well‐posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time‐splitting scheme, one can naturally build unconditionally stable semi‐implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin‐film growth process. Copyright © 2017 John Wiley & Sons, Ltd.     

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015