Hybrid solvers for composition and splitting methods

Composition and splitting are useful techniques for constructing special purpose integration methods for numerically solving many types of differential equations. In this article we will review these methods and summarise the essential ingredients of an implementation that has recently been added to a framework for solving differential equations in Mathematica.     

Comparing time integrators for parabolic equations in two space dimensions with a mixed derivative

A numerical comparison is made between three integration methods for semi-discrete parabolic partial differential equations in two space variables with a mixed derivative. Linear as well as non-linear equations are considered. The integration methods are the well-known one-step line hopscotch method, a four-step line hopscotch method, and a stabilized, explicit Runge-Kutta method.     

A comparative study of ADI splitting methods for parabolic equations in two space dimensions

The main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

Solving the Dirichlet problem for Navier–Stokes equations by probabilistic approach

We construct a number of layer methods for Navier-Stokes equations (NSEs) with no-slip boundary conditions. The methods are obtained using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the proposed methods are nevertheless deterministic.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

On Time Discretizations for Spectral Methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.     

Differential equations driven by rough paths with jumps

We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.     

On Nonlinear Galerkin Approximation

1.IntroductionInordertosolvetheproblemoflongtimeilltegrationofevolutionpartialdifferentialequations,nPnlinearGalerkinmethodsareintroducedinrecentyears.SuchmethodsstemfromthetheoryofinertialmanifoldsandaPprokimateinertialmanifolds.Werecallaninertialmanifoldisafinitedimensionalsmoothmanifoldwhichcontainstheglobalattractora-ndattractseveryorbitatanexponelltial.ate[1'2].However,therearestillmanydissipativepartialdifferelltialequationsforwhichtheexistenceofinertialmanifoldsisnotknown;thereareeven…     

A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions

In this paper, we propose a new numerical algorithm for solving linear and non linear fractional differential equations based on our newly constructed integer order and fractional order generalized hat functions operational matrices of integration. The linear and nonlinear fractional order differential equations are transformed into a system of algebraic equations by these matrices and these algebraic equations are solved through known computational methods. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm. The results obtained, using the scheme presented here, are in full agreement with the analytical solutions and numerical results presented elsewhere.     

求解非线性Black-Scholes模型的自适应小波精细积分法

针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性.     

The control of order and steplength for backward differentiation methods

Backward differentiation methods are used extensively for integration of stiff systems of ordinary differential equations, but most implementations are inefficient when some of the eigenvalues of the Jacobi matrix are close to the imaginary axis. For these problems the performance of backward differentiation methods can be improved considerably by application of the instability test and reaction which is described in this paper. During instability the local truncation error oscillates rapidly with increasing magnitude. This property is used in the instability test. When instability is detected the order is lowered as much as possible without reducing the steplength.The instability test and reaction is derived from a simplified analysis of integration of linear systems of differential equations, and the performance is verified for a number of linear test problems.This work was supported by »Statens Teknisk-Videnskabelige Forskningsråd« under grant no. 516-6537. E-368.     

On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root

Summary A class of linear multistep methods is considered for the numerical integration of stiff systems of ordinary differential equations. These methods are characterized by the fact that the coefficients of the integration formulas are matrices depending on the Jacobian or on an approximation to the Jacobian. They have the possibility to adapt the characteristic root of the method to the problem under consideration. Special attention is paid to stability aspects. Numerical results are reported.     

Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.     

Symplectic phase flow approximation for the numerical integration of canonical systems

Summary New methods are presented for the numerical integration of ordinary differential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincaré invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a Runge-Kutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Hénon-Heiles system, which describes the motion of a star in an axisymmetric galaxy.     

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are di?cult to mimic using conventional time integrators that are used to solve systems of ordinary di?erential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.     

多体系统动力学微分／代数方程组的一类新的数值分析方法

本文讨论了多体系统动力学微分／代数混合方程组的数值离散问题．首先把参数t并入广义坐标讨论，简化了方程组及其隐含条件的结构，并将其化为指标1的方程组．然后利用方程组的特殊结构，引入一种局部离散技巧并构造了相应的算法．算法结构紧凑，易于编程，具有较高的计算效率和良好的数值性态，且其形式适合于各种数值积分方法的的实施．文末给出了具体算例．     

Comparing the contractivity properties of semi-implicit methods

Contractivity is a desirable property of numerical integration methods for stiff systems of ordinary differential equations. In this paper, numerical parameters are used to allow a direct and quantitative comparison of the contractivity properties of various methods for non-linear stiff problems. Results are provided for popular Rosenbrock methods and some more recently developed semi-implicit methods.     

Efficient transient noise analysis in circuit simulation

Stochastic differential algebraic equations (SDAEs) arise as a mathematical model for electrical network equations that are influenced by additional sources of Gaussian white noise. We discuss adaptive linear multi-step methods for their numerical integration, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, and we obtain conditions that ensure mean-square convergence of this methods. For the case of small noise we present a strategy for controlling the step-size in the numerical integration. It is based on estimating the mean-square local errors and leads to step-size sequences that are identical for all computed paths. Test results illustrate the performance of the presented methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)