Semidiscretization In Time Research Articles

Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation

We consider a time semi-discretization of a generalized Allen–Cahn equation with time-step parameter $$\tau $$ . For every $$\tau $$ , we build an exponential attractor $$\mathcal {M}_\tau $$ of the discrete-in-time dynamical system. We prove that $$\mathcal {M}_\tau $$ converges to an exponential attractor $$\mathcal {M}_0$$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $$\tau $$ tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $$\mathcal {M}_\tau $$ is bounded by a constant independent of $$\tau $$ . Our construction is based on the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semi-group. Their result has been applied in many situations, but here, for the first time, the perturbation is a discretization. Our method is applicable to a large class of dissipative problems.

The 1D Richards’ equation in two layered soils: a Filippov approach to treat discontinuities

Abstract The infiltration process into the soil is generally modeled by the Richards’ partial differential equation (PDE). In this paper a new approach for modeling the infiltration process through the interface of two different soils is proposed, where the interface is seen as a discontinuity surface defined by suitable state variables. Thus, the original 1D Richards’ PDE, enriched by a particular choice of the boundary conditions, is first approximated by means of a time semidiscretization, that is by means of the transversal method of lines (TMOL). In such a way a sequence of discontinuous initial value problems, described by a sequence of second order differential systems in the space variable, is derived. Then, Filippov theory on discontinuous dynamical systems may be applied in order to study the relevant dynamics of the problem. The numerical integration of the semidiscretized differential system will be performed by using a one-step method, which employs an event driven procedure to locate the discontinuity surface and to adequately change the vector field.

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $$W^{1,\infty }$$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain $$W^{1,\infty }$$ estimates, which are the key to the numerical analysis of these problems.

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type

Semidiscretization in time is studied for a class of quasi-linear evolution equations in a framework due to Kato, which applies to symmetric first-order hyperbolic systems and to a variety of fluid and wave equations. In the regime where the solution is sufficiently regular, we show stability and optimal-order convergence of the linearly implicit and fully implicit midpoint rules and of higher-order implicit Runge–Kutta methods that are algebraically stable and coercive, such as the collocation methods at Gauss nodes.

An Error Analysis of a Finite Element Method with IMEX-Time Semidiscretizations for Some Partial Integro-differential Inequalities Arising in the Pricing of American Options

The aim of this article is to develop and analyze a finite element method, combined with implicit-explicit (IMEX) time semidiscretizations, for pricing American options under Merton's and Kou's jum...

Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge–Kutta methods for Hamiltonian semilinear evolution equations

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

Homogenization of degenerate coupled fluid flows and heat transport through porous media

We establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media. Existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time, a priori estimates and passing to the limit from discrete approximations. After that, porous material exhibiting periodic spatial oscillations is considered and the two-scale convergence (as the oscillation period vanishes) to a corresponding homogenized problem is rigorously proven.

Discrete maximal parabolic regularity for Galerkin finite element methods

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, for example, for establishing optimal a priori error estimates in non- Hilbertian norms without unnatural coupling of spatial mesh sizes with time steps.

Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation

We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated withdiscretization is nonincreasing at every time iteration. We prove that the sequence generated by the scheme converges to a steady state as time tends to infinity. We also obtain convergence rates in the energy norm. The proof is based on the Łojasiewicz-Simon inequality.

Runge-Kutta time semidiscretizations of semilinear PDEs with non-smooth data.

We study semilinear evolution equations frac{mathrm dU}{mathrm dt}=AU+B(U) posed on a Hilbert space mathcal Y, where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from mathcal Y_ell = D(A^ell ) to itself, and ell in I subseteq [0,L], L ge 0, 0,L in I. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge–Kutta method in time, retaining continuous space, and prove convergence of order O(h^{pell /(p+1)}) for non-smooth initial data U^0in mathcal Y_ell , where ell le p+1, for a method of classical order p, extending a result by Brenner and Thomée for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.

On a nonlinear partial differential algebraic system arising in the technical textile industry: analysis and numerics

In this paper we explore a numerical scheme for a nonlinear fourth order system of partial differential algebraic equations that describes the dynamics of slender inextensible elastica as they arise in the technical textile industry. Applying a semi-discretization in time, the resulting sequence of nonlinear elliptic systems with the algebraic constraint for the local length preservation is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is based on a finite element discretization in space. The simulation results confirm the analytically predicted properties of the scheme.

Weak solutions of coupled dual porosity flows in fractured rock mass and structured porous media

This paper deals with a fully nonlinear degenerate parabolic system with natural (critical) growths and under non-linear boundary conditions. Such problems arise from the heat and water flow through a partially saturated fractured rock mass and structured porous media. Existence of a global weak solution of the problem (on an arbitrary interval of time) is proved by means of semidiscretization in time, deriving suitable a-priori estimates based on W1,p-regularity of the approximate solution and by passing to the limit from discrete approximations.

On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt γ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.

On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δtγ) for any γ < ½. We also prove that the scheme converges uniformly in the strong Lp-sense but with no rate given.

Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models

In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of $$O(MNlog_{2}(M))$$O(MNlog2(M)) for Merton model and of $$O(MN)$$O(MN) for Kou model, where $$N$$N denotes the number of time steps and $$M$$M the number of collocation points. Some numerical examples, for pricing European options under Merton and Kou jump-diffusion models with constant as well as variable volatility, are presented to demonstrate the stability, convergence and computational complexity of the methods.

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.

Crank--Nicolson Time Stepping and Variational Discretization of Control-Constrained Parabolic Optimal Control Problems

We consider a control-constrained parabolic optimal control problem and use variational discretization for its time semidiscretization. The state equation is treated with a Petrov--Galerkin scheme using a piecewise constant Ansatz for the state and piecewise linear continuous test functions. This results in variants of the Crank--Nicolson scheme for the state and the adjoint state. Exploiting a superconvergence result, we prove second order convergence in time of the error in the controls. Moreover, the piecewise linear and continuous parabolic projection of the discrete state on the dual time grid provides a second order convergent approximation of the optimal state without further numerical effort. Numerical experiments confirm our analytical findings.

On the existence of solutions for a drift-diffusion system arising in corrosion modeling

In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.

An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation

This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative noises. Within this framework, the proposed scheme is shown to converge uniformly in the strong $L^p$-sense with precise convergence rates given. The abstract results are then applied to several concrete examples. Further, weak convergence rates of the scheme are examined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques based on the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions on the nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus. The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Both strong and weak convergence results obtained here show that the proposed method achieves higher convergence rates than the implicit Euler and Crank-Nicolson time discretizations. Numerical results are finally reported to confirm our theoretical findings.

Analysis of a drift–diffusion model with velocity saturation for spin-polarized transport in semiconductors

A system of drift–diffusion equations with electric field under Dirichlet boundary conditions is analyzed. The system of strongly coupled parabolic equations for particle density and spin density vector describes the spin-polarized semi-classical electron transport in ferromagnetic semiconductors. The presence of a nonconstant and nonsmooth magnetization vector, solution of the Landau–Lifshitz equation, causes the diffusion matrix to be dependent on space and time and to have in general poor regularity properties, thus making the analysis challenging. To partially overcome the analytical difficulties the velocity saturation hypothesis is made, which results in a bounded drift velocity. The global-in-time existence and uniqueness of weak solutions is shown by means of a semi-discretization in time, which yields an elliptic semilinear problem, and a quadratic entropy inequality, which allow for the limit of vanishing time step size. The convergence of the weak solutions to the steady state, under some restrictions on the parameters and data, is shown. Finally, the higher regularity of solutions for a smooth magnetization in two space dimensions is shown through a diagonalization argument, which allows to get rid of the cross diffusion terms in the fluid equations, and the iterative application of Gagliardo–Nirenberg inequalities and a generalized version of Aubin lemma.