Runge-Kutta Collocation Methods Research Articles

[formula omitted]- and [formula omitted]-stability of Runge–Kutta collocation methods

This paper deals with stability of classical Runge–Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [13] and used in [14] for the time integration of nonlinear evolution PDEs, the stability of these methods has to be adapted to this context. For this reason, we develop in this paper several notions of stability, that we analyze. We provide sufficient conditions that can be checked algorithmically using just the collocation points of the method to ensure that these stability notions are fulfilled by a given Runge–Kutta collocation method. We also introduce examples and counterexamples used in [14] to highlight the necessity of these stability conditions in this context.

A Clas of A-Stable Runge-Kutta Collocation Methods for the Solution of First Order Ordinary Dierential Equations

This paper presented a class of A-stable Runge-Kutta collocation methods with three free parameters for the solution of rst order ordinary dierential equations. Power series was considered as its basis function, adoption of interpolation and collocation of the approximate solution at some selected grid points to give system of equations was also considered. Gaussian Elimination method was used to solve for the unknown parameters and substituted into the approximate solution to give the continuous method. The three cases considered are the Guass, the Lobatto, and the Radau types. Analysis of the methods was made based on order, zero stability, consistence and convergence. The derived schemes were implemented in the Predictor-Corrector mode. Comparison with existing methods showed that the new developed Schemes compete favorably.

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Second derivative Runge-Kutta collocation methods for the numerical solution of sti system of rst order initial value problems in ordinary dierential equations are derived and studied. The inclusion of the second derivative terms enabled us to derive a set of methods which are convergent with large regions of absolute stability. Although the implementation of the methods remains iterative in a precisely dened way, the advantage gained makes them suitable for solving sti system of equations with large Lipschitz constants. The derived methods are illustrated by the applications to some test problems of sti system found in the literature and the numerical results obtained conrm the potential of the second derivative methods.

Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems

We introduce a new class of implicit two-derivative Runge–Kutta collocation methods designed for the numerical solution of systems of equations and show how they have been implemented in an efficient parallel computing environment. We also discuss the difficulty associated with large systems and how, in this case, one must take advantage of the second derivative terms in the methods. We consider two modified versions of the methods which are suitable for solving stable systems. The first modification involves the introduction of collocation at the two end points of the integration interval in addition to the Gaussian interior collocation points and the second involves the introduction of a different class of basic second derivative methods. With these modifications, fewer function evaluations per step are achieved, resulting into methods that are cheap and easy to implement. The stability properties of these methods are investigated and numerical results are given for each of the modified version to illustrate the computational efficiency of the modified methods.

On the global error of special Runge–Kutta methods applied to linear Differential Algebraic Equations

Global error estimates are obtained for Runge–Kutta methods of special type when applied to linear constant coefficient Differential Algebraic Equations (DAEs) of arbitrary high index ν ≥ 0 . A Runge–Kutta formula is said of special type when its first internal stage is computed explicitly, the remaining internal stages are obtained in terms of a regular coefficient submatrix whereas the last internal stage equals the advancing solution. As a main result, one extra order of convergence on arbitrary high index ν ≥ 2 linear constant coefficient DAEs is obtained for a one parameter family of strictly stable Runge–Kutta collocation methods of special type when compared to the classical Radau IIA formulae for the same number of implicit stages.

Strongly [formula omitted]-stable first stage explicit collocation methods with stepsize control for stiff and differential–algebraic equations

A variable stepsize implementation of a recently introduced one-parameter family of high order strongly A-stable Runge–Kutta collocation methods with the first internal stage of explicit type is presented. The so-called SAFERK(α,s) methods with free parameter α and s stages are well-suited for the integration of stiff and differential–algebraic systems, and they are computationally equivalent to the (s−1)-stage Radau IIA method, since they all have a similar amount of implicitness. For the same number of implicit stages, both SAFERK(α,s) and Radau IIA(s−1) methods possess algebraic order 2s−3, whereas the stage order is one unit higher for SAFERK methods. Although there are no L-stable schemes in this method family, the free parameter α can be selected in order to minimize the error coefficients or to maximize the numerical dissipation. Besides a general discussion of the method class, it is shown here how the 4-stage methods can be endowed with an embedded third order formula, and an implementation based on the perfected RADAU5 code with an adaptive stepsize controller proves to be competitive for a wide selection of test problems including electric circuit analysis, constrained mechanical systems, and time-dependent partial differential equations treated by the method of lines.

Runge-Kutta collocation methods for differential-algebraic equations of indices 2 and 3

Stiffly accurate Runge-Kutta collocation methods with explicit first stage are examined. The parameters of these methods are chosen so as to minimize the errors in the solutions to differential-algebraic equations of indices 2 and 3. This construction results in methods for solving such equations that are superior to the available Runge-Kutta methods.

An efficient family of strongly A-stable Runge–Kutta collocation methods for stiff systems and DAEs. Part II: Convergence results

The convergence properties of a newly defined uniparametric family of collocation Runge–Kutta methods on non-stiff systems, stiff semi-linear problems and Differential-Algebraic Equations are analysed. For each s⩾3, the so-called family of SAFERK methods is based on interpolatory quadrature rules of order 2s−3, and they have the same number of implicit stages as the s-stage LobattoIIIA and (s−1)-stage RadauIIA methods. Since s-stage SAFERK methods possess the same algebraic order as the (s−1)-stage RadauIIA method, the corresponding principal terms of local error are compared, and it is shown that those SAFERK methods with positive weights and nodes located in the integration interval possess, in the l2-norm, a smaller principal term of local error. Moreover, it is proved that SAFERK methods are convergent when integrating relevant classes of stiff problems and Differential-Algebraic Equations. In particular, they possess a higher stiff order than the RadauIIA method of the same algebraic order, due to a higher stage order of the new methods. On the other hand, because of the strong A-stability, SAFERK methods can be competitive with LobattoIIIA methods, particularly for stiff problems on large time intervals.

Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge–Kutta collocation methods

The construction of a class of backward differentiation formulae based on intra-step Chebyshev–Gauss–Lobatto nodal points, suitable for the approximate numerical integration of initial-value problems of first-order ordinary differential equations, is presented. Formulae of this new family are A 0-stable and L(α)-stable for any orders, and, particularly, for orders 1 and 2 they are L-stable. Regions of absolute stability and stability measures make this class very promising. We prove that this family of methods may be considered as Runge–Kutta collocation methods where the abscissae are obtained from the Chebyshev–Gauss–Lobatto points.

Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems

An error analysis for a newly defined uniparametric family of stiffly accurate Runge-Kutta collocation methods when applied to initial value problems for singularly perturbed differential equations is carried out. The so-called SAFERK methods possess a first internal stage of explicit type and are based on collocation nodes. Sharp convergence results are obtained for these methods through the analysis of a sequence of higher index Differential Algebraic Equations. A numerical test with the Van der Pol oscillator reveals that the proposed error estimates are realistic whenever the stepsize h is large enough compared to the stiffness parameter e.

An efficient family of strongly [formula omitted]-stable Runge–Kutta collocation methods for stiff systems and DAEs. Part I: Stability and order results

For each integer s ≥ 3 , a new uniparametric family of stiffly accurate, strongly A -stable, s -stage Runge–Kutta methods is obtained. These are collocation methods with a first internal stage of explicit type. The methods are based on interpolatory quadrature rules, with precision degree equal to 2 s − 4 , and all of them have two prefixed nodes, c 1 = 0 and c s = 1 . The amount of implicitness of our s -stage method is similar to that involved with the s -stage LobattoIIIA method or with the ( s − 1 ) -stage RadauIIA method. The new family of Runge–Kutta methods proves to be of interest for the numerical integration of stiff systems and Differential Algebraic Equations. In fact, on several stiff test problems taken from the current literature, two methods selected in our 4-stage family, seem to be slightly more efficient than the 3 -stage RadauIIA method and also more robust than the 4 -stage LobattoIIIA method.

On the relations between B 2V Ms and Runge–Kutta collocation methods

The principal aim of this paper is a rigorous analysis of the relations between Block Boundary Value Methods (B 2V Ms) with minimal blocksize defined over a suitable nonuniform finer mesh and well-known Runge–Kutta collocation methods. Moreover, a further aspect that will be briefly investigated is the construction of an extended finer mesh for building B 2V Ms with nonminimal blocksize. Some advantages that may arise from the use of the so-obtained methods will be also discussed.

A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

We consider the construction of a special family of Runge–Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev–Gauss–Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and the fourth-order Runge–Kutta–Lobatto IIIA method. We will present here the coefficients up to eighth order, but we provide the formulas to obtain methods of higher order. When the number of stages is odd, we have considered a new strategy for changing the step size based on the use of a pair of methods: the given RK method and a linear multistep one. Some numerical experiments are considered in order to check the behaviour of the methods when applied to a variety of initial-value problems.

Exponential fitted Runge–Kutta methods of collocation type: fixed or variable knot points?

Two different classes of exponential fitted Runge–Kutta collocation methods are considered: methods with fixed points and methods with frequency-dependent points. For both cases we have obtained extensions of the classical two-stage Gauss, RadauIIA and LobattoIIIA methods. Numerical examples reveal important differences between both approaches.

Review Paper: Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations

In recent years, after a prolonged period of intense activity, the study of numerical methods for solving stiff initial-value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity. There now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems. In this paper, we sketch some of the main theory which underpins stiff integration methods and we use this to describe, and put into context, some of the best codes currently available. By referencing only codes which have been thoroughly tested and are widely available, our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type. An additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated.

Multi-Symplectic Runge–Kutta Collocation Methods for Hamiltonian Wave Equations

A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss–Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for Hamiltonian ODEs. We also discuss the issue of conservation of energy and momentum. Since time discretization by a Gauss–Legendre method is computational rather expensive, we suggest several semi-explicit multi-symplectic methods based on Gauss–Legendre collocation in space and explicit or linearly implicit symplectic discretizations in time.

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.

Runge-Kutta collocation methods for rigid body lie-poisson equations

The rigid body Lie-Poisson structure in three dimensions is considered. We show that the symplectic collocation type Runge-Kutta methods preserve the one-form of the underlying system. The linear error growth, energy and momentum conservation properties of the numerical solutions are discussed for Euler top equation.

On theA 0-acceptability of rational approximations to the exponential function with only real poles

We consider rational approximations to the exponential function with real poles, γ1−1,...,γm−1, that correspond to implicit Runge-Kutta collocation methods. We show that if γi≥1/2,i=1,...,m, the rational approximation isA0-acceptable.