In this paper stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the $l^\infty(0,T;H)$-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The $l^\infty(0,T;V)$-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.
We consider applying the Strang splitting to semilinear parabolic problems. The key ingredients of the Strang splitting are the decomposition of the equation into several parts and the computation of approximate solutions by combining the time evolution of each split equation. However, when the Dirichlet boundary condition is imposed, order reduction could occur due to the incompatibility of the split equations with the boundary condition. In this paper, to overcome the order reduction, a modified Strang splitting procedure is presented for the one-dimensional semilinear parabolic equation wit…
This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.
A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Investigation on numerical schemes of these equations with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of sem…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfull…
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the autho…
Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs
Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP（波城大学、E2S 巴约讷-比亚里茨大学、法国国家科学研究中心、数学与及其应用实验室）；Univ Rennes, INRIA (MINGuS), IRMAR - UMR 6625, ENS Rennes（雷恩大学、法国国家信息与自动化研究所（MINGuS）、雷恩数学研究所（UMR 6625）、雷恩高等师范学院）；Univ Rennes, CNRS, IRMAR - UMR 6625, ENS Rennes（雷恩大学、法国国家科学研究中心、雷恩数学研究所（UMR 6625）、雷恩高等师范学院）；Section of Mathematics, University of Geneva（日内瓦大学数学系）
We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac …
A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this problem, we employ the method of upper and lower solutions to obtain sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. In particular, those results imply that milder (compared to the optimal) grading yields the optimal convergence rate $2-α$ in positive time, while quasi-uniform temporal meshes yield first-orde…
This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the …
This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel's formula, first- and second-ord…
This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the backward Euler. We consider a larger class of nonlinear drift functions, which are of Nemytskii type and polynomial of any odd degree with negative leading term, instead of only dealing with the special case of stochastic Allen-Chan equation as in the up to date literature.Moreover our linear operator is of second order and not necessarily self-adjoint, therefo…
A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have $k$th-order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data.
The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t=\mathcal{L}u+f[u]$ with $\mathcal{L}$ being a linear dissipative operator and $f$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its init…
The Strang splitting method, formally of order two, can suffer from order reduction when applied to semilinear parabolic problems with inhomogeneous boundary conditions. The recent work [L .Einkemmer and A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 1. Dirichlet boundary conditions. SIAM J. Sci. Comput., 37, 2015. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., 38, 2016] introduces a modification of the method to avoid the reduction of order based on the nonlinearity. In this paper we introduce a new correction constructed directly from the flow of t…
We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $ε$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $Δt$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $ε$, in terms of $Δt$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in …
For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semi…
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
