The implementation of Runge-Kutta implicit processes

The application ofA-stable, implicit Runge-Kutta processes to the solution of stiff systems of ordinary differential equations is discussed, and an iterative procedure for solving the resulting nonlinear system of equations is suggested.     

Efficient higher order implicit one-step methods for integration of stiff differential equations

A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.     

On the D-suitability of implicit Runge-Kutta methods

The concept of suitability means that the nonlinear equations to be solved in an implicit Runga-Kutta method have a unique solution. In this paper, we introduce the concept of D-suitability and show that previous results become special cases of ours. In addition, we also give some examples to illustrate the D-suitability of a matrixA.     

Error bounds for the solution to the algebraic equations in Runge-Kutta methods

In the implementation of implicit Runge-Kutta methods inaccuracies are introduced due to the solution of the implicit equations. It is shown that these errors can be bounded independently of the stiffness of the differential equation considered if a certain condition is satisfied. This condition is also sufficient for the existence and uniqueness of a solution to the algebraic equations. TheBSI-andBS-stability properties of several classes of implicit methods are established.     

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.     

Parallel iterative linear solvers for multistep Runge-Kutta methods

This paper deals with solving stiff systems of differential equations by implicit Multistep Runge-Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge-Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. With a parallel iterative linear system solver, especially designed for MRK methods, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-steps-stage MRK applied with this technique is on s processors equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. A simple implementation of both methods shows that, for the same number of Newton iterations, the accuracy delivered by the new method is higher than that of BDF.     

On the efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A ^T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.     

A-stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.This author was partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998     

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.     

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi??s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s?1 in the singly-implicit case with arbitrary step sizes and s??5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.     

Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics

We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.     

Structural properties and decomposition of linear balanced matrices

Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column.The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi. In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orA ^T contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.Partial support from NSF grant DMS 8606188, ECS 8800281 and DDM 8800281.     

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smith?s methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.     

An inverse eigenvalue problem: ComputingB-stable Runge-Kutta methods having real poles

The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.     

Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of Hamiltonian ordinary differential equations by means of Newton-like iterations. We pay particular attention to time-symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For an s-stage IRK scheme used to integrate a \(\dim \)-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same \(s\dim \times s\dim \) real coefficient matrix. We propose a technique that takes advantage of the symplecticity of the IRK scheme to reduce the cost of methods based on diagonalization of the IRK coefficient matrix. This is achieved by rewriting one step of the method centered at the midpoint on the integration subinterval and observing that the resulting coefficient matrix becomes similar to a skew-symmetric matrix. In addition, we propose a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation.     

A local-global version of a stepsize control for Runge-Kutta methods

In this paper we develop a new procedure to control stepsize for Runge-Kutta methods applied to both ordinary differential equations and semi-explicit index 1 differential-algebraic equations. In contrast to the standard approach, the error control mechanism presented here is based on monitoring and controlling both the local and global errors of Runge-Kutta formulas. As a result, Runge-Kutta methods with the local-global stepsize control solve differential or differential-algebraic equations with any prescribed accuracy (up to round-off errors). For implicit Runge-Kutta formulas we give the sufficient number of both full and modified Newton iterations allowing the iterative approximations to be correctly used in the procedure of the local-global stepsize control. In addition, we develop a stable local-global error control mechanism which is applicable for stiff problems. Numerical tests support the theoretical results of the paper.     

广义中立型系统的渐近稳定性及数值分析

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