Asynchronous Parareal Time Discretization For Partial Differential Equations

In this note, we discuss strong convergence of exponential integrator scheme based on spatial and time discretization for a class of neutral stochastic partial differential equations driven by α-stable processes.

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A Numerical Method of High Accuracy for Linear Parabolic Partial Differential Equations

Mathematical Problems in Engineering ◽

10.1155/2012/497365 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14

Author(s):

Jun Liu ◽

Yan Wang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Computational Cost ◽

Time Discretization ◽

Parabolic Partial Differential Equations ◽

Spline Collocation ◽

Partial Differential ◽

Quadratic Spline ◽

Space Partition ◽

Gauss Method

We report a new numerical algorithm for solving one-dimensional linear parabolic partial differential equations (PDEs). The algorithm employs optimal quadratic spline collocation (QSC) for the space discretization and two-stage Gauss method for the time discretization. The new algorithm results in errors of fourth order at the gridpoints of both the space partition and the time partition, and large time steps are allowed to save computational cost. The stability of the new algorithm is analyzed for a model problem. Numerical experiments are carried out to confirm the theoretical order of accuracy and demonstrate the effectiveness of the new algorithm.

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Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2008.0325 ◽

2008 ◽

Vol 465 (2102) ◽

pp. 649-667 ◽

Cited By ~ 63

Author(s):

Arnulf Jentzen ◽

Peter E Kloeden

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Numerical Approximation ◽

Time Discretization ◽

Space Time ◽

Euler Scheme ◽

Numerical Schemes ◽

Partial Differential ◽

The Laplace Operator

We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

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Numerical Approximation of Stationary Distributions for Stochastic Partial Differential Equations

Journal of Applied Probability ◽

10.1239/jap/1409932678 ◽

2014 ◽

Vol 51 (3) ◽

pp. 858-873 ◽

Cited By ~ 4

Author(s):

Jianhai Bao ◽

Chenggui Yuan

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stationary Distribution ◽

Stochastic Partial Differential Equations ◽

Numerical Approximation ◽

Time Discretization ◽

Weak Limit ◽

Step Size ◽

Partial Differential ◽

Exponential Integrator

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.

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Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

Abstract and Applied Analysis ◽

10.1155/2013/857205 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Jingjun Zhao ◽

Jingyu Xiao ◽

Yang Xu

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Partial Differential Equations ◽

Differential Equations ◽

Time Discretization ◽

Stability And Convergence ◽

Fractional Partial Differential Equations ◽

Weak Formulation ◽

Partial Differential ◽

Element Method

A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.

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MATHEMATICAL ANALYSIS AND NUMERICAL METHODS FOR A PARTIAL DIFFERENTIAL EQUATIONS MODEL GOVERNING A RATCHET CAP PRICING IN THE LIBOR MARKET MODEL

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005465 ◽

2011 ◽

Vol 21 (07) ◽

pp. 1479-1498 ◽

Cited By ~ 5

Author(s):

A. PASCUCCI ◽

M. SUÁREZ-TABOADA ◽

C. VÁZQUEZ

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Interest Rates ◽

Time Discretization ◽

Fundamental Solutions ◽

Market Model ◽

Computational Time ◽

Cauchy Problems ◽

Partial Differential ◽

Libor Market Model

In this paper, we present a mathematical model for pricing a particular financial product: the ratchet cap. This derivative product depends on certain interest rates (whose dynamics we assume that follow the LIBOR market model). The pricing model is rigorously posed in terms of a sequence of nested Cauchy problems associated to uniformly parabolic partial differential equations. First, for each problem the existence and uniqueness of solution is obtained. Next, this analysis allows to propose a new and more efficient numerical method based on the approximation by computable fundamental solutions of constant coefficient operators. The advantage in terms of computational time of this new modeling and analytically based approach is illustrated by comparison with the classically used Monte Carlo simulation and a characteristics Crank–Nicolson time discretization combined with finite elements strategy.

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