Trigonometric fitted,eighth‐order explicit Numerov‐type methods

We consider the integration of the special second‐order initial value problem of the form . A recently introduced family of 7 stages, eighth‐order methods, sharing constant coefficients, is used as base. This family is properly modified to derive phase fitted and zero dissipative methods (ie, trigonometric fitted) that are best suited for integrating oscillatory problems. Numerical tests over a set of problems shows enhanced performance when the purely linear part of the problems is rather large in comparison with the rest of nonlinear parts. An appendix implementing a MATLAB listing with the coefficients of the new method is also given.     

B-convergence of the implicit midpoint rule and the trapezoidal rule

We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.1980 AMS Subject Classification: 65L05, 65L20.     

Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems

As shown in part I of this paper and references therein, the classical method of Iterated Defect Correction (IDeC) can be modified in several nontrivial ways, extending the flexibility and range of applications of this approach. The essential point is an adequate definition of the defect, resulting in a significantly more robust convergence behavior of the IDeC iteration, in particular, for nonequidistant grids. The present part II is devoted to the efficient high-order integration of stiff initial value problems. By means of model problem investigation and systematic numerical experiments with a set of stiff test problems, our new versions of defect correction are systematically evaluated, and further algorithmic measures are proposed for the stiff case. The performance of the different variants under consideration is compared, and it is shown how strong coupling between non-stiff and stiff components can be successfully handled. AMS subject classification 65L05 Supported by the Austrian Research Fund (FWF) grant P-15030.     

Diagonally implicit Runge-Kutta methods for stiff problems

Diagonally implicit Runge-Kutta methods are examined. It is shown that, for stiff problems, the methods based on the minimization of certain error functions have advantages over other methods; these functions are determined in terms of the errors for simplest model equations. Methods of orders three, four, five, and six are considered.     

On the iterative solution of the algebraic equations in fully implicit Runge-Kutta methods

This paper deals with the iterative solution of stage equations which arise when some fully implicit Runge-Kutta methods, in particular those based on Gauss, Radau and Lobatto points, are applied to stiff ordinary differential equations. The error behaviour in the iterates generated by Newton-type and, particularly, by single-Newton schemes which are proposed for the solution of stage equations is studied. We consider stiff systems y'(t) = f(t,y(t)) which are dissipative with respect to a scalar product and satisfy a condition on the relative variation of the Jacobian of f(t,y) with respect to y, similar to the condition considered by van Dorsselaer and Spijker in [7] and [17]. We prove new convergence results for the single-Newton iteration and derive estimates of the iteration error that are independent of the stiffness. Finally, some numerical experiments which confirm the theoretical results are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Phase‐fitted,six‐step methods for solving x′′ = f(t,x)

An explicit, six‐step method of sixth order is presented and tuned for the numerical solution of x′′ = f(t,x). This method is explicit, hybrid, and uses two function evaluations (stages) per step. Its coefficients are varied and depend on the step size. This variance comes from the demand of the method to nullify the phase errors produced when solving the standard simple oscillator. The first and second derivative of this error vanish also. Numerical tests in a set of relevant problems illustrate the efficiency of the newly derived method.     

On the Starting Algorithms for Fully Implicit Runge-Kutta Methods

This paper is concerned with the behavior of starting algorithms to solve the algebraic equations of stages arising when fully implicit Runge-Kutta methods are applied to stiff initial value problems. The classical Lagrange extrapolation of the internal stages of the preceding step and some variants thereof that do not require any additional cost are analyzed. To study the order of the starting algorithms we consider three different approaches. First we analyze the classical order through the theory of Butcher's series, second we derive the order on the Prothero and Robinson model and finally we study the stiff order for a general class of dissipative problems. A detailed study of the orders of some starting algorithms for Gauss, Radau IA-IIA, Lobatto IIIA-C methods is also carried out. Finally, to compare the most relevant starting algorithms studied here, some numerical experiments on well known nonlinear stiff problems are presented.     

Fitted modifications of classical Runge‐Kutta pairs of orders 5(4)

Runge‐Kutta pairs sharing orders 5 and 4 are among the most celebrated methods for solving initial value problems. Here, after considering the phase lag as a function of frequency v, we derive a modification of a particular pair. Namely, we present a zero dissipative pair with vanished phase error and its first derivative. After extended numerical tests in various oscillatory problems, it seems that this modified pair outperforms existing methods in the literature.     

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Peer two-step W-methods are designed for integration of stiff initial value problems with parallelism across the method. The essential feature is that in each time step s ‘peer’ approximations are employed having similar properties. In fact, no primary solution variable is distinguished. Parallel implementation of these stages is easy since information from one previous time step is used only and the different linear systems may be solved simultaneously. This paper introduces a subclass having order s−1 where optimal damping for stiff problems is obtained by using different system parameters in different stages. Favourable properties of this subclass are uniform stability for realistic stepsize sequences and a superconvergence property which is proved using a polynomial collocation formulation. Numerical tests on a shared memory computer of a matrix-free implementation with Krylov methods are included. AMS subject classification (2000) 65L06, 65Y05.Received June 2004. Revised January 2005. Communicated by Timo Eirola.Helmut Podhaisky: The work of this author was supported by the German Academic Exchange Service, DAAD.     

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

     

On the Convergence of Backward Differentiation Formulas for Stiff Initial Value Problems

The influence of a time-dependent transformation to a numerical method is studied. Thus convergence results of backward differentiation formulas applied to the non-autonomous stiff system y = A(t)y + (t) are given. The approach is based on a special decomposition of the companion matrix.This revised version was published online in October 2005 with corrections to the Cover Date.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Exponential fitted Gauss, Radau and Lobatto methods of low order

Several exponential fitting Runge-Kutta methods of collocation type are derived as a generalization of the Gauss, Radau and Lobatto traditional methods of two steps. The new methods are capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. A different procedure to find the parameter of the method is proposed. The variable step Radau method of two stages is derived. Finally, numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.      

New splitting methods for two-dimensional evolutionary equations

New second- and third-order splitting methods are proposed for evolutionary-type partial differential equations in a two-dimensional space. These methods are derived on the basis of diagonally implicit methods applied to the numerical analysis of stiff ordinary differential equations. The splitting methods are found to be absolutely unconditionally stable. Test calculations are presented.     

Generalized solutions of parabolic initial value problems with discontinuous nonlinearities

We proved recently that parabolic initial value problems with discontinuous nonlinearities have no unique weak solution in general, but have a unique generalized solution in the sense of Colombeau. In this paper we study the relationship between generalized solutions and weak solutions.     

Trigonometric–fitted hybrid four–step methods of sixth order for solving

A two–stage, explicit, hybrid four–step method of sixth order for the solution of the special second order initial value problem is presented here. The new method is trigonometric fitted, thus it uses variable coefficients. Numerical tests illustrate the superiority of our proposal over similar methods found in the relevant literature on a set of standard problems.     

The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients

In this article, higher dimensional initial boundary value problems of variable coefficients are solved by means of an analytic technique, namely the Homotopy analysis method (HAM). Comparisons are made between the Adomian decomposition method (ADM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Generalized linear elasticity in exterior domains. I: Radiation problems

We use Hodge–Helmholtz decompositions of weighted Sobolev spaces to solve time-harmonic exterior-boundary value problems for perturbations of the (aδd+bdδ)-system (δ: the co-differential, a, b>0). We prove, that a Fredholm alternative holds true, the eigensolutions decay polynomially at infinity, and that the positive eigenvalues do not accumulate. © 1997 B. G. Teubner Stuttgart-John Wiley & Sons Ltd.     

一类二阶常微分方程初值问题的无穷多解性

应用分离变量法,得到了一类二阶微分方程初值问题存在无穷多个非负解的充分必要条件,并给出了所有的无穷多个非负解．。     

New fourth-order splitting methods for two-dimensional evolution equations

New second- and third-order splitting methods are proposed for partial differential equations of the evolution type in a two-dimensional space. The methods are derived as based on diagonal implicit techniques used in the numerical solution to stiff ordinary differential equations. The methods are absolutely and unconditionally stable. Test computations are presented.