Multistep Time Discretizations Research Articles

Solving BSDEs based on novel multi-step schemes and multilevel Monte Carlo

For backward stochastic differential equations (BSDEs), we construct a fully explicit multi-step time-discretization scheme and prove its stability and convergence rates. To approximate conditional expectations in our scheme, we design a new algorithm based on the multilevel Monte Carlo (MLMC) method. This estimator can achieve a prescribed error ϵ with a computational effort of order ϵ−2. Numerical experiments are given to illustrate the theoretical results of the proposed methods.

A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps

We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose a novel way to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations.

Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation

We derive some a posteriori error estimates for the Richards equation. This parabolic equation is nonlinear in space and in time, thus its resolution requires fixed-point iterations within each time step. We measure the approximation error with the dual norm of the residual. A computable upper bound of this error consists of several estimators involving adequate reconstructions based on the degrees of freedom of the scheme. The space and time reconstructions are specified for a two-step backward differentiation formula and a discrete duality finite volume scheme. Our strategy to decrease the computational cost relies on an aggregation of the estimators in three components: space discretization, time discretization, and linearization. We propose an algorithm to stop the fixed-point iterations after the linearization error becomes negligible, and to choose the time step in order to balance the time and space errors. We analyze the influence of the parameters of this algorithm on three test cases and quantify the gain obtained in comparison with a classical simulation.

A third-order multistep time discretization for a Chebyshev tau spectral method

A time discretization scheme based on the third-order backward difference formula has been embedded into a Chebyshev tau spectral method for the Navier-Stokes equations. The time discretization is a variant of the second-order backward scheme proposed by Krasnov et al. (2008) 3. High-resolution direct numerical simulations of turbulent incompressible channel flow have been performed to compare the backward scheme to the Runge-Kutta scheme proposed by Spalart et al. (1991) 2. It is shown that the Runge-Kutta scheme leads to a poor convergence of some third-order spatial derivatives in the direct vicinity of the wall, derivatives that represent the diffusion of wall-tangential vorticity. The convergence at the wall is shown to be significantly improved if the backward scheme is applied.

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

New one‐leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non‐negativity and the entropy dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G‐stability theory with entropy dissipation methods and to introduce a nonlinear transformation of variables, which provides a quadratic structure in the equations. It is shown that G‐stability of the one‐leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross‐diffusion system from population dynamics and a nonlinear fourth‐order quantum diffusion model, for which the existence of semidiscrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second‐order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1119–1149, 2015

A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our s

Weighted [formula omitted] Paley–Wiener Theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces

We study the weighted l1 Paley–Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra equation of scalar type in a Hilbert space. The results extend to weighted l1 stability some recent l1 stability theorem due to the author (Numer. Math. 109 (2008) 571–595). Our theoretical results are applied to investigate the asymptotic behavior as tn→∞ of the numerical solutions of certain integro-differential equations in Hilbert space.

On the Linear Stability of the Fifth-Order WENO Discretization

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.

Multirate Explicit Adams Methods for Time Integration of Conservation Laws

This paper constructs multirate linear multistep time discretizations based on Adams-Bashforth methods. These methods are aimed at solving conservation laws and allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps—restricted by the largest value of the Courant number on the grid—and therefore results in more efficient computations. Numerical results obtained for the advection and Burgers’ equations confirm the theoretical findings.

A higher-order generalized ghost fluid method for the poor for the three-dimensional two-phase flow computation of underwater implosions

The ghost fluid method for the poor (GFMP) is an elegant, computationally efficient, and nearly conservative method for the solution of two-phase flow problems. It was developed in one dimension for the stiffened gas equation of state (EOS) and one-step time-discretization algorithms. It naturally extends to three dimensions but its extension to higher-order, multi-step time-discretization schemes is not straightforward. Furthermore, the original GFMP and many other ghost fluid methods fail to handle the large density and pressure jumps that are encountered in underwater implosions. Therefore, the GFMP is generalized in this work to an arbitrary EOS and multi-fluid problems with multiple EOSs. It is also extended to three dimensions and developed for higher-order, multi-step time-discretization algorithms. Furthermore, this method is equipped with an exact two-phase Riemann solver for computing the fluxes across the material interface without crossing it. This aspect of the computation is a departure from the standard approach for computing fluxes in ghost fluid methods. It addresses the stiff nature of the two-phase air/water problem and enables a better handling of the large discontinuity of the density at the air/water interface. As the original GFMP, the proposed method is contact preserving, computationally efficient, and nearly conservative. Its superior performance in the presence of large density and pressure jumps is demonstrated for shock-tube problems. Its practicality and accuracy are also highlighted with the three-dimensional simulation of the implosion of an air-filled and submerged glass sphere.

Implicit–explicit methods based on strong stability preserving multistep time discretizations

In this note we propose and analyze novel implicit–explicit methods based on second order strong stability preserving multistep time discretizations. Several schemes are developed, and a linear stability analysis is performed to study their properties with respect to the implicit and explicit eigenvalues. One of the proposed schemes is found to have very good stability properties, with implicit A-stability for the entire explicit stability domain. The properties of the other proposed schemes are comparable to those of traditional methods found in the literature.

Convergence of Multistep Time Discretizations of Nonlinear Dissipative Evolution Equations

Global error bounds are derived for multistep time discretizations of fully nonlinear evolution equations on infinite dimensional spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic Lipschitz constants and nonlinear semigroups. The error bounds reveal how the contractive or dissipative behavior of the vector field, governing the evolution, and the properties of the multistep method influence the convergence. A multistep method which is consistent of order $p$ is proven to be convergent of the same order when the vector field is contractive or strictly dissipative, i.e., of the same order as in the ODE-setting. In the contractive context it is sufficient to require strong zero-stability of the method, whereas strong $A$-stability is sufficient in the dissipative case.

On high order strong stability preserving runge-kutta and multi step time discretizations

On high order strong stability preserving runge-kutta and multi step time discretizations

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties---in any norm or seminorm--of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge---Kutta methods for nonlinear problems and for linear problems as well as implicit Runge---Kutta methods and multi step methods will be collected

Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations

Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected (Ascher et al., 1995). In this paper, we develop Runge-Kutta-based IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range.

On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations

Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data.