Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme

Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.

     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

A capacitance matrix method for Dirichlet problem on polygon region

Summary An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN ²log₂ N (N=1/h) where the cost of solving the capacitance matrix equations isNlog₂ N on regular grids andN ^3/2log₂ N on irregular ones.     

Norm bounds for rational matrix functions

Summary If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then f(A)₂1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

Extrapolation at stiff differential equations

Summary Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied tostiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour ofextrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included.     

Stability of ZakianI MN recursions for linear delay differential equations

Long sequences of linear delay differential equations (DDEs) frequently occur in the design of control systems with delays using iterative-numerical methods, such as the method of inequalities. ZakianI _MN recursions for DDEs are suitable for solving this class of problems, since they are reliable and provide results to the desired accuracy, economically even if the systems are stiff. This paper investigates the numerical stability property of theI _MN recursions with respect to Barwell's concept ofP-stability. The result shows that the recursions using full gradeI _MN approximants areP-stable if, and only if,N−2≤M≤N−1.     

A study of Rosenbrock-type methods of high order

Summary This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.     

A 0-Stability and stiff stability of Brown's multistep multiderivative methods

Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA ₀-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA ₀-stable. It is seen that for allk andl withk1.5 (l+1) the methods areA ₀-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A ₀-stability has been determined numerically by implementing the necessary and sufficient condition.     

Projektive Newton-Verfahren bei elliptischen Randwertaufgaben

Summary In this paper we investigate projective Newton methods for nonlinear elliptic boundary value problems. These methods yield approximations by solving the linear equations of the Newton method by a projection method, e.g. the Ritz method. Using subspaces of finite elements or polynominals we obtain error estimates and optimal convergence theorems (inH ₁ andL ₂).

     

B-convergence properties of defect correction methods. I

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that.hL is small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

Two-step numerical methods for parabolic differential equations

A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order     

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     

Error of rosenbrock methods for stiff problems studied via differential algebraic equations

This paper studies Rosenbrock methods when they are applied to stiff differential equations containing a small stiffness parameter. The basic ideas and techniques are the same as those developed for Runge-Kutta methods in an earlier paper of the authors. The results obtained here are essentially those obtained for Diagonally Implicit Runge-Kutta methods.     

Die maximale Konsistenzordnung von Differenzenapproximationen nichtnegativer Art

Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.

     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.