A class of rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations

AbstractWe develop a class of generalized Rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations. We convert the second-order systems to equivalent first-order form, and then employ the square of the Jacobian. These methods when applied to a linear time-invariant system Uu + AU = 0, reproduce a class of schemes given by Baker and Bramble that are derived from a particular class of rational approximations to the exponential with denominators of the form (1 − γ2z2)s for an s-stage method. For our problem, then, an s-stage scheme requires the solution of 2s linear algebraic systems at each time step, with the same real matrix. We employ the theory of Butcher [1–4] series to develop order conditions and then present specific examples of fourth-order methods which are unconditionally stable by appropriate choice of parameter γ2. Numerical results, confirming the rate of convergence, are presented

Compact schemes in time with applications to partial differential equations

We propose a new class of fourth-and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrodinger equation, wave equation, Burgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods.S. Clain and G.J. Machado acknowledge the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. M.T. Malheiro acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. S. Clain, G.J. Machado, and M.T. Malheiro acknowledge the fi-nancial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 - Programa Operacional Fatores de Competitividade, POCI-01-0145-FEDER-028118 and PTDC/MAT-APL/28118/2017

Multistep-multistage-multiderivative methods for ordinary differential equations

This paper studies a general method for the numerical integration of ordinary differential equations. The method, defined in part 1, contains many known processes as special case, such as multistep methods, Runge-Kutta methods (multistage), Taylor, series (multiderivative) and their extensions (section 2). After a short section on trees and pairs of trees we derive formulas for the conditions to be satisfied by the free parameters in order to equalize the numerical approximation with the solution up to a certain order. Next we extend the reuslts of Kastlunger [6]. The proof given here is shorter than the original one. Finally we discuss formulas, with the help of which the conditions for the parameters can be reduced considerably and give numerical examples