Two-step peer methods with equation-dependent coefficients

We introduce a new class of explicit two-step peer methods with the aim of improving the stability properties of already existing peer methods, by making use of coefficients depending on the Jacobian of the Ordinary Differential Equations (ODEs) system to solve. Numerical tests highlight the best stability and accuracy properties of the new methods compared to the classical and equation-dependent ones proposed in Conte et al. (Lect Notes Comput Sci 12949:309-324, 2021)

Two classes of linearly implicit numerical methods for stiff problems: analysis and MATLAB software

The purpose of this work lies in the writing of efficient and optimized Matlab codes to implement two classes of promising linearly implicit numerical schemes that can be used to accurately and stably solve stiff Ordinary Differential Equations (ODEs), and also Partial Differential Equations (PDEs) through the Method Of Lines (MOL). Such classes of methods are the Runge-Kutta (RK) [28] and the Peer [17], and have been constructed using a variant of the Exponential-Fitting (EF) technique [27]. We carry out numerical tests to compare the two methods with each other, and also with the well known and very used Gaussian RK method, by the point of view of stability, accuracy and computational cost, in order to show their convenience

Diagonally implicit exponentially fitted Runge-Kutta methods with equation dependent coefficients

It is the purpose of this paper to derive diagonally implicit exponentially fitted methods for the numerical solution of initial value problems based on first order ordinary differential equations. The approach used takes into account the contribution to the error originated from the computation of the internal stages approximations. The derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages, as classically done in the classical derivation of multistage EF-based methods (compare [3] and references therein). Standard and revised EF methods are then compared in terms of linear stability and numerical performances

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods