Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

     

Convergence Analysis of Spatially Adaptive Rothe Methods

This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by \(S\) -stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances in each time step must be tuned in order to preserve the asymptotic temporal convergence order of the time stepping also in the presence of spatial discretization errors. In particular, we discuss the case of linearly implicit time integrators and adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance.     

Some results on the stability and dynamics of finite difference approximations to nonlinear partial differential equations

A miscellany of results on the nonlinear instability and dynamics of finite difference discretizations of the Burgers and Kortweg de Vries equations is obtained using a variety of phase-plane, functional analytic, and regularity methods. For the semidiscrete (space-discrete, time-continuous) schemes, large-wave-numer instabilities occurring in special exact solutions are investigated, and parameter values for which the semidiscrete scheme is monotone are considered. For fully discrete schemes (space and time discrete), large-wave-number instabilities introduced by various time-stepping schemes such as forward Euler, leapfrog, and Runge–Kutta schemes are analyzed. Also, a time step restriction for the monotonicity of the forward-Euler time-stepping scheme, and regularity of a 4-stage monotone/conservative Runge–Kutta time stepping are investigated. The techniques used here may be employed, in conjunction with bifurcation-theoretic and weakly nonlinear analyses, to analyze the stability of numerical schemes for other nonlinear partial differential equations of both dissipative and dispersive varieties. © 1993 John Wiley & Sons, Inc.     

Linearly implicit methods for nonlinear evolution equations

Summary.  We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta schemes and extrapolation. We establish optimal order error estimates. The abstract results are applied to a third–order evolution equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method. The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels. Received October 22, 2001 / Revised version received April 22, 2002 / Published online December 13, 2002 Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06 Correspondence to: G. Akrivis     

Numerical study of the transverse stability of the Peregrine solution

We generalize a previously published numerical approach for the one-dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth-order method for the time integration, based on a splitting scheme and an implicit Runge-Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a y-independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.     

非线性Galerkin算法的收敛性和复杂性

本文给出了数值求解非线性发展方程的全离散非线性Ｇａｌｅｒｋｉｎ算法,即将空间离散时的谱非线性Ｇａｌｅｒｋｉｎ算法和时间离散的Ｅｕｌｅｒ差分格式相结合,得到了显式和隐式两种全离散数值格式,相应地也考虑了显式和隐式的Ｇａｌｅｒｋｉｎ全离散格式,并分别分析了上述四种全离散格式的收敛性和复杂性,经过比较得出结论;在某些约束条件下,非线性Ｇａｌｅｒｋｉｎ算法和Ｇａｌｅｒｋｉｎ算法具有相同阶的收敛速度,然而前     

High-order time integration methods in molecular dynamics

The mechanical behaviour of molecular structures can be described with stiff differential equations, which can not be solved analytically. Several numerical time integration schemes can be found in the literature. The aim of this paper is to present the class of partitioned Runge-Kutta methods applied in molecular dynamics. This class of methods includes a wide range of explicit and implicit, single- and multi-stage, symplectic and non-symplectic, low- and high-order time integration schemes. Also most of the classical methods like explicit and implicit Euler, explicit and implicit midpoint rule, Störmer-Verlet and Newmark are also partitioned Runge-Kutta methods. The schemes are implemented in a finite element code which can serve as a numerical platform for molecular dynamics. This code is used to show the sensitivity of the simulations to the accuracy of the initial values. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linearly implicit time discretization for free surface problems

Deeper investigation of time discretization for free surface problems is a widely neglected problem. Many existing approaches use an explicit decoupling which is only conditionally stable. Only few unconditionally stable methods are known, and known methods may suffer from too strong numerical dissipativity. They are also usually of first rder only [1, 9]. We are therefore looking for unconditionally stable, minimally dissipative methods of higher order. Linearly implicit Runge-Kutta (LIRK) methods are a class of one-step methods that require the solution of linear systems in each time step of a nonlinear system. They are well suited for discretized PDEs, e.g. parabolic problems [7]. They have been used successfully to solve the incompressible Navier-Stokes equations [5]. We suggest an adaption of these methods for free surface problems and compare different approximations to the Jacobian matrix needed for such methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-phase laminar axisymmetric jet flow: Explicit,implicit, and split-operator approximations

An hybrid Eulerian-Lagrangian numerical scheme is developed for a two-phase problem and four finite-difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second-order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas-phase equations has been solved by four different numerical methods: a predictor-corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator-splitting method. The subsystem of liquid-phase equations is solved along the droplet trajectories by a second-order Runge-Kutta scheme. The computations have been made to predict the hydro-dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference-schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.     

A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows

In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.     

A note on the efficient implementation of implicit methods for ODEs

The use of implicit methods for ODEs, e.g. implicit Runge-Kutta schemes, requires the solution of nonlinear systems of algebraic equations of dimension s · m, where m is the size of the continuous differential problem to be approximated. Usually, the solution of this system represents the most time-consuming section in the implementation of such methods. Consequently, the efficient solution of this section would improve their performance. In this paper, we propose a new iterative procedure to solve such equations on sequential computers.     

Mono-implicit Runge-Kutta schemes for the parallel solution of initial value ODEs

Among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit Runge-Kutta (IRK) schemes. The calculation of the stages of the IRK method involves the solution of a nonlinear system of equations usually employing some variant of Newton's method. Since the costs of the linear algebra associated with the implementation of Newton's method generally dominate the overall cost of the computation, many subclasses of IRK schemes, such as diagonally implicit Runge-Kutta schemes, singly implicit Runge-Kutta schemes, and mono-implicit (MIRK) schemes, have been developed to attempt to reduce these costs. In this paper we are concerned with the design of MIRK schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors. This work builds on that of an earlier investigation in which a special subclass of the MIRK formulas were implemented in parallel. While suitable parallelism was achieved, the formulas were limited to some extent because they all had only stage order 1. This is of some concern since in the application of a Runge-Kutta method to a system of stiff ODEs the phenomenon of order reduction can arise; the IRK method can behave as if its order were only its stage order (or its stage order plus one), regardless of its classical order. The formulas derived in the current paper represent an improvement over the previous investigation in that the full class of MIRK formulas is considered and therefore it is possible to derive efficient, parallel formulas of orders 2, 3, and 4, having stage orders 2 or 3.     

On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.     

Second-order time-integration in inelastic dynamical systems

In structural dynamics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization two classes of Runge-Kutta methods and the generalized-α method are compared. The representatives for the Runge-Kutta methods are diagonally implicit Runge-Kutta schemes (DIRK) and diagonally implicit Runge-Kutta-Nyström schemes (DIRKN). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. AMS subject classification (2000) 65M30, 65M15, 35K60     

Global optimization of explicit strong-stability-preserving Runge-Kutta methods

Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (specifically, for fifth-order methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature.

     

Second-order accurate projective integrators for multiscale problems

We introduce new projective versions of second-order accurate Runge–Kutta and Adams–Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer–inner projective integrators to be applied to legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.     

A comparison of time adaptive integration methods for small and large strain viscoelasticity

In quasistatic solid mechanics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization three different classes of Runge-Kutta methods are compared. These methods are diagonally implicit Runge-Kutta schemes (DIRK), linear implicit Runge-Kutta methods (Rosenbrock type methods) and half-explicit Runge-Kutta schemes (HERK). It will be shown that the application of half-explicit or linear-implicit Runge-Kutta methods can enormously reduce the computational time in particular situations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constraint Preserving,Inexact Solution of Implicit Discretizations of Landau–Lifshitz–Gilbert Equations and Consequences for Convergence

The Landau-Lifshitz-Gilbert equation describes dynamics of ferromagnetism. Nonlinearity of the equation and a non-convex side constraint make it difficult to design reliable approximation schemes. In this paper, we discuss the numerical solution of nonlinear systems of equations resulting from implicit, unconditionally convergent discretizations of the problem. Numerical experiments indicate that finite-time blow-up of weak solutions can occur and thereby underline the necessity of the design of reliable discretization schemes that approximate weak solutions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

Summary We address the question of convergence of fully discrete Runge-Kutta approximations. We prove that, under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time will be strictly smaller than the conventional one, a phenomenon called order reduction. This phenomenon is extensively discussed, both analytically and numerically. As distinct from earlier contributions we here treat explicit Runge-Kutta schemes. Although our results are valid for both parabolic and hyperbolic problems, the examples we present are therefore taken from the hyperbolic field, as it is in this area that explicit discretizations are most appealing.