We study the numerical integration of large stiff systems of differential equations by methods that use matrix--vector products with the exponential or a related function of the Jacobian. For large problems, these can be approximated by Krylov subspace methods, which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators. The exponential methods also offer favorable properties in the integration of differential equations whose Jacobian has large imaginary eigenvalues. We derive methods up to order 4 which are exact for linear constant-coefficient equations. The implementation of the methods is discussed. Numerical experiments with reaction-diffusion problems and a time-dependent Schrodinger equation are included.

/pdf/exponential-integrators-for-large-systems-of-differential-4caoai1lvf.pdf

Exponential Integrators for Large Systems of Differential Equations

Analysis and implementation of TR-BDF2

Haar wavelet approach to nonlinear stiff systems

/pdf/mathematical-modeling-of-catalytic-fixed-bed-reactors-45vp366ehy.pdf

Mathematical modeling of catalytic fixed bed reactors

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 5. A finite-difference, adaptive space/time grid strategy based on a patch-type local uniform spatial grid refinement, for kinetic models in one-dimensional space geometry

We study the effect of transfer of phase noise in different four wave mixing schemes using a coherent phase noise measurement technique. The nature of phase noise transfer from the pump to the generated wavelengths is shown to be independent of the type of phase noise (1 / f or white noise frequency components). We then propose a novel scheme using dual correlated pumps to prevent the increase in phase noise in the conjugate wavelengths. The proposed scheme is experimentally verified by the all-optical wavelength conversion of a DQPSK signal at 10.7 GBaud.

Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion.

We present a convergence analysis for a class of one-step, exponentially fitted, finite difference schemes for stiff initial-value problems. Such schemes, when applied to the numerical integration of the linear scalar problem ϵy ′ + a ( x ) y = f ( x ), x ∈ (0, X ), with y (0) given and where ϵ > 0 is a small parameter, give solutions satisfying a uniform (in ϵ) error estimate. In this paper, a set of sufficient conditions for uniform second-order convergence is derived. The findings differ from those reported in [1–3], and are complemented by the results of numerical experiments which illustrate the effectiveness of the proposed approach.

Sufficient conditions for uniformly second-order convergent schemes for stiff initial-value problems

A composite integration scheme for the numerical solution of systems of parabolic PDEs in one space dimension

We examine a number of adaptive space mesh routines which were designed to provide accurate numerical approximations to the solutions of parabolic partial differential equations in one space dimension. Both static and dynamic regridding techniques are considered, and in numerical experiments, we compare their relative performances by measuring a range of statistics including accuracy and cpu when applied to approximate the solutions of a set of representative test problems.

A comparison of some adaptive space mesh solvers for the numerical solution of parabolic partial differential equations

We present a new class of one-step, multi-value Exponential Integrator (EI) methods referred to as Exponential Almost Runge-Kutta (EARK) methods which involve the derivatives of a nonlinear function of the solution. In order to approximate such derivatives to a sufficient accuracy, the EARK methods will be implemented within the broader framework of Exponential Almost General Linear Methods (EAGLMs) to accommodate past values of this nonlinear function and becoming multistep in nature as a consequence. Established EI methods, such as Exponential Time Differencing (ETD) methods, Exponential Runge-Kutta (ERK) methods and Exponential General Linear Methods (EGLMs) become special cases of EAGLMs. We present order conditions which facilitate the construction of two- and three-stage EARK methods and, when cast in an EAGLM format, we perform a stability analysis to enable a comparison with existing EI methods. We conclude with some numerical experiments which confirm the convergence order and also demonstrate the computational efficiency of these new methods.

Exponential almost Runge-Kutta methods for semilinear problems

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