Implementation Of Implicit Runge-Kutta Methods Research Articles

Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge–Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler (the simplest implicit RK method) and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution where the system is solved as a whole and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into a number of smaller systems using the Jordan normal form of the RK matrix and then solve them successively.For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that the conditioning of an implicit RK method behaves like that of the implicit Euler method. If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite.The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Illustrative numerical examples are given.

A CLASS OF S-STEP NON-LINEAR ITERATION SCHEME BASED ON PROJECTION METHOD FOR GAUSS METHOD

Various iteration schemes are proposed by various authors to solve nonlinear equations arising in the implementation of implicit Runge-Kutta methods. In this paper, a class of s-step non-linear scheme based on projection method is proposed to accelerate the convergence rate of those linear iteration schemes. In this scheme, sequence of numerical solutions is updated after each sub-step is completed. For 2-stage Gauss method, upper bound for the spectral radius of its iteration matrix was obtained in the left half complex plane. This result is extended to 3-stage and 4-stage Gauss methods by transforming the coefficient matrix and the iteration matrix to a block diagonal form. Finally, some numerical experiments are carried out to confirm the obtained theoretical results.

Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method

Various iteration schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. A modified Newton scheme is typically used to solve these equations. As an alternative to this scheme, iteration schemes, which sacrifice super-linear convergence for reduced linear algebra costs, have been proposed. A scheme of this type proposed in [11] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [11] when it is applied to the scalar test differential equation 􀢞′ 􀵌 􀢗􀢞 and the convergence rate depends on the spectral radius of the iteration matrix 􀡹􁈺􀢠􁈻, a function of 􀢠 􀵌 􀢎􀢗, where 􀢎 is the step-length. In this scheme, some conditions are imposed on the spectral radius of its iteration matrix in order to get super-linear convergence at two points in the 􀢠 -plane. Then the supremum of the spectral radius of each scheme is minimized over the left-half 􀢠 -plane in order to improve the convergence rate of the scheme. Two new schemes with optimal parameters are obtained for the two-stage Gauss method and some numerical experiments are reported.

An efficient scheme for the implementation of implicit Runge-Kutta methods

A scheme is proposed for solving nonlinear algebraic equations arising in the implementation of the implicit Runge-Kutta methods. In contrast to the available schemes, not only the starting values of the variables but also those of the derivatives are predicted. This makes it possible to reduce the number of evaluations of the function (the right-hand side) at each implicit stage without significantly reducing the accuracy of integration.

Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

AbstractVarious iterative schemes have been proposed to solve the non‐linear equations arising in the implementation of implicit Runge‐Kutta methods. In one scheme, when applied to an s‐stage Runge‐Kutta method, each step of the iteration still requires s function evaluations but consists of r(>s) sub‐steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, …, and superlinear convergence is obtained in the limit k→∞. Some results are obtained for Gauss methods and numerical results are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Runge-Kutta methods: some historical notes

This centenary history of Runge-Kutta methods contains an appreciation of the early work of Runge, Heun, Kutta, and Nyström and a survey of some significant developments of these methods over the last hundred years. In particular, the order conditions, as they are now understood, will be outlined, as will the introduction and practical implementation of implicit Runge-Kutta methods, the use of linear and nonlinear stability analysis in the assessment of Runge-Kutta methods, and the theory and applications of the composition of methods. Of the many further developments that have arisen, most are discussed only briefly; however, the recent interest in Runge-Kutta methods for Hamiltonian systems will be surveyed in more detail.

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix.

On the use of parallel processors for implicit Runge-Kutta methods

An iteration scheme, for solving the non-linear equations arising in the implementation of implicit Runge-Kutta methods, is proposed. This scheme is particularly suitable for parallel computation and can be applied to any method which has a coefficient matrixA with all eigenvalues real (and positive). For such methods, the efficiency of a modified Newton scheme may often be improved by the use of a similarity transformation ofA but, even when this is the case, the proposed scheme can have advantages for parallel computation. Numerical results illustrate this. The new scheme converges in a finite number of iterations when applied to linear systems of differential equations, achieving this by using the nilpotency of a strictly lower triangular matrixS−1AS — Λ, with Λ a diagonal matrix. The scheme reduces to the modified Newton scheme whenS−1AS is diagonal.A convergence result is obtained which is applicable to nonlinear stiff systems.

Some schemes for the implementation of implicit Runge-Kutta methods

The nonlinear equations, arising in the implementation of implicit Runge-Kutta methods, may be solved by a modified Newton iteration, but alternative iteration schemes have been suggested to reduce the linear algebra costs. A linear iteration scheme is examined in this article. When applied to an s-stage Runge-Kutta method, each step of the iteration requires s function evaluations and the solution of s sets of linear equations. For the scalar differential equation x′ = qx, the convergence rate of the scheme depends on the spectral radius π[ M( z)] of the iteration matrix M, a function of z = hq where h is the steplength. A lower bound for π[ M( z)] is established and new schemes are obtained for the two-stage Gauss method by minimizing the supremum of this lower bound over regions of the complex plane. In one scheme the supremum on the negative real axis is minimized. The iteration scheme is generalized in order to obtain improved convergence rates. When applied to an s-stage Runge-Kutta method, each step of this new scheme still requires just s function evaluations. However r sets of linear equations, r > s, have to be solved in each step. Some results are obtained for the Gauss methods and some numerical experiments reported.

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.

A scheme for the implementation of implicit Runge-Kutta methods

The computational work required to implement implicit Runge-Kutta methods is often dominated by the cost of solving large sets of nonlinear equations. As an alternative to modified Newton methods, iteration schemes, which sacrifice superlinear convergence for reduced linear algebra costs, have been proposed. A new scheme of this type is considered here. This scheme avoids expensive vector transformations, is computationally more efficient, and gives improved performance.

On the implementation of implicit Runge-Kutta methods

The modified Newton iterations in the implementation of ans stage implicit Runge-Kutta method for ann dimensional differential equation system require 2s3n3/3+O(n2) operations for theLU factorisations and 2s2n2+O(n) operations for the back substitutions. This paper describes a method for transforming the linear system so as to reduce these operation counts.