In this work, we introduce the class of self starting general linear methods that are multi-stage multi-step methods, which do not require any additional starting and finishing procedures. We provide sufficient conditions to derive methods with Runge-Kutta stability and show how some properties of diagonally implicit Runge-Kutta methods can be improved, keeping similar computational costs.

Explicit general linear method of two-step peer-type of order 1 to 4 suitable for solving non-stiff ordinary differential equations are derived. Estimates of the local truncation error are derived with which a variable stepsize and variable order scheme is implemented. The new methods are compared in numerical experiments with ode45 from Matlab.

A new class of G(ϵ)-symplectic general linear methods for numerical integration of Hamiltonian systems of differential equations is described. Order conditions for these methods are derived using Albrecht approach and the construction of G(ϵ)-symplectic method is described based on solving minimization problems with nonlinear inequality constrains. Examples of methods up to the order four are presented. Numerical experiments confirm that these methods achieve the expected order of accuracy and that they approximately preserve Hamiltonians of differential systems.

We describe the construction of G- or G(ϵ)-symplectic, and parasitism free or ϵ-parasitism free general linear methods for numerical integration of Hamiltonian systems of differential equations. Examples of such methods are presented up to the order p=4 and stage order q=p−1. Numerical experiments confirm that all methods achieve the expected order of accuracy, and that these methods approximately preserve Hamiltonians as well as quadratic invariants of differential systems.

This paper reviews strong stability preserving discrete variable methods for differential systems. The strong stability preserving Runge–Kutta methods have been usually investigated in the literature on the subject, using the so-called Shu–Osher representation of these methods, as a convex combination of first-order steps by forward Euler method. In this paper, we revisit the analysis of strong stability preserving Runge–Kutta methods by reformulating these methods as a subclass of general linear methods for ordinary differential equations, and then using a characterization of monotone general linear methods, which was derived by Spijker in his seminal paper (SIAM J Numer Anal 45:1226–1245, 2007). Using this new approach, explicit and implicit strong stability preserving Runge–Kutta methods up to the order four are derived. These methods are equivalent to explicit and implicit RK methods obtained using Shu–Osher or generalized Shu–Osher representation. We also investigate strong stability preserving linear multistep methods using again monotonicity theory of Spijker.

We systematically investigate strong stability preserving general linear methods of order p, stage order q=p or q=p−1, with r=p+1 external approximations, and s=p−1 internal approximations, for numerical solution of differential systems. Examples of methods of order p and stage order q=p or q=p−1, with large strong stability preserving coefficients and large regions of absolute stability, are provided for p=2, p=3, and p=4. The results of numerical experiments confirm that the methods constructed in this paper achieve the expected order of accuracy, do not produce spurious oscillations, and they are suitable to preserve the monotonicity of the numerical solution, when applied to discretization of hyperbolic conservation laws with discontinuous initial conditions.

Explicit two-step Runge-Kutta (TSRK) methods offer an efficient alternative to traditional explicit Low-Storage Runge-Kutta (LSRK) schemes for solving the Navier-Stokes equations. A special class of TSRK methods that reduce requirement compared to previous TSRK schemes are derived. Schemes of fourth, fifth and sixth order are implemented and tested. The new schemes are evaluated with two common test cases, a 2D cylinder and a 3D Taylor-Green vortex. The results are compared with classical time discretization strategies. Timings obtained in three different hardware configurations show that the new TSRK methods of order four are 25% faster than LSRK schemes of the same order. Fifth and sixth order TSRK methods are tested with the same 3D test case and the results are compared to LSRK algorithms. Results show TSRK schemes of fifth and sixth order are competitive compared to LSRK methods of the same orders, as LSRK methods are of second order for non linear differential systems. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

We describe a new approach to the construction of rational functions of high dispersive orders, and SDIRK methods with dispersive stability functions, for the numerical solution of differential systems with oscillatory solutions. The numerical experiments on test problems with periodic or almost periodic solutions confirm the order and dispersive order of convergence of the proposed numerical schemes.

Computational fluid dynamics (CFD) has emerged as a successful tool for industry applications and basic science during the last decades. However, accurate solutions involving vortex propagation, and separated and turbulent flows, are still associated with high computing costs. In particular, large eddy simulations (LES) of complex geometries, such as a complete automobile, require several days on thousands of cores in order to obtain solutions with statistically relevant information. With an increase in the number of available cores, the number of degrees of freedom (DOF) per core can be reduced accordingly. When the number of DOF per core is below a certain threshold the total simulation time is not bounded by floating point operations (FLOPS), but by the time spend on communication between cores. To overcome this impediment we have identified and tested a class of two-step Runge-Kutta (TSRK) methods of high order with low number of stages, for time discretization of differential systems resulting from space discretization of weakly compressible Navier-Stokes equations. These methods have not been used before in CFD simulations. The advantage of using these methods is reduction in communication times between cores. The numerical experiments indicate that the gains in computational performance of this new class of TSRK methods, as compared with classical Runge-Kutta (RK) methods or low storage Runge-Kutta (LSRK) schemes, are of the order of25%, with no loss in accuracy.

We consider the class of implicit–explicit (IMEX) general linear methods (GLMs) to construct methods where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has inherent Runge–Kutta stability (IRKS) property, and it is A-, or L-stable. We will also investigate the absolute stability of these methods when the implicit and explicit parts interact with each other. In particular, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is A(α)-stable for α∈[0,π∕2]. Finally we furnish examples of SSP IMEX GLMs up to the order p=4 and stage order q=p with optimal SSP coefficients.

A mathematical model is formulated for the fox rabies epidemic that swept through large areas of Europe during parts of the last century. Differently from other models, both territorial and diffusing rabid foxes are included, which leads to a system of partial differential, functional differential and differential-integral equations. The system is reduced to a scalar Volterra-Hammerstein integral equation to which the theory of spreading speeds pioneered by Aronson and Weinberger is applied. The spreading speed is given by an implicit formula which involves the space-time Laplace transform of the integral kernel. This formula can be exploited to find the dependence of the spreading speed on the model ingredients, in particular on those describing the interplay between diffusing and territorial rabid foxes and on the distribution of the latent period.

We describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.

We describe some issues related to the development of a new code for numerical solution of systems of Volterra integral equations of the second kind. This code is based on A- and V0-stable natural Volterra Runge-Kutta method of order p=4 implemented in a variable stepsize environment. This method was derived by Conte et al. (2014) [13]. The numerical experiments on many equations and systems of equations illustrate that the code achieves the expected order of accuracy, and confirm its efficiency and robustness.

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by an implicit formula, and the non-stiff part is integrated by an explicit formula. We will construct methods where the explicit part has strong stability preserving (SSP) property, and the implicit part of the method is A-, or L-stable. We will also investigate stability of these methods when the implicit and explicit parts interact with each other. To be more precise, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is A-, or L-stable. Finally, we furnish examples of SSP IMEX DIMSIMs up to the order four with good stability properties.

Various issues are discussed which are relevant to the development of the code for nonstiff differential systems based on a class of general linear methods with inherent Runge–Kutta stability.

We extend our previous work on the spatial spread of fox rabies from one dimension to two dimensions. We consider the case when the latent period has fixed length. We use the method of lines to replace the spatial derivatives and the integral equations with algebraic approximations, then we apply the explicit continuous Runge–Kutta method of fourth order and discrete Runge–Kutta method of third order with six stages to numerically integrate the resulting systems of ordinary and delay differential equations. We discuss and confirm some of the major results we obtained in earlier work. The asymptotic speeds of spread observed in the two-dimensional simulations and in earlier work are discussed and compared with those found in nature.

In this paper we investigate the strong stability preserving (SSP) property of transformed diagonally implicit multistage integration methods (DIMSIMs). Within this class, examples of SSP methods of order p=1,2,3 and 4 and stage order q=p have been determined, and also suitable starting and finishing procedure have been constructed. The numerical experiments performed on a set of test problems have shown that SSP transformed DIMSIMs can be more accurate and competitive with SSP Runge–Kutta methods of the same order.

We investigate strong stability preserving (SSP) general linear methods (GLMs) for systems of ordinary differential equations. Such methods are obtained by the solution of the minimization problems with nonlinear inequality constrains, corresponding to the SSP property of these methods, and equality constrains, corresponding to order and stage order conditions. These minimization problems were solved by the sequential quadratic programming algorithm implemented in MATLAB\(^{\circledR }\) subroutine fmincon.m starting with many random guesses. Examples of transformed SSP GLMs of order \(p = 1, 2, 3\), and 4, and stage order \(q = p\) have been determined, and suitable starting and finishing procedures have been constructed. The numerical experiments performed on a set of test problems have shown that transformed SSP GLMs constructed in this paper are more accurate than transformed SSP DIMSIMs and SSP Runge–Kutta methods of the same order.