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In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

Runge–Kutta embedded pairs of high algebraic order are frequently utilized when strict tolerances are required. When creating such pairings of orders nine and eight for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients' magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders 9 and 8 with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems. [ABSTRACT FROM AUTHOR]

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli (Numer. Algo., 27, 119–130 2021). In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the modified extended second derivative backward differentiation formulae from second derivative general linear methods point of view. This makes it possible to open some maneuver rooms in developing the methods with superior features by perturbing the abscissa vector of the methods. The proposed methods are constructed to have better accuracy and stability properties in comparison with the original ones. These improvements are verified by giving some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we describe an approach to estimate the global error for explicit second derivative general linear methods based on the approach which has been already used for global error estimation of explicit general linear methods. In this approach, to estimate the global error, we use the numerical solutions of pairs of second derivative general linear methods with the same order and stage order that are constructed such that their global error functions are proportional. Numerical experiments demonstrate the excellent agreement of the global error estimation with the exact one in both constant and variable stepsize environments. [ABSTRACT FROM AUTHOR]

In this paper, we exploit the Strang splitting technique for solving the multidimensional Allen-Cahn equations with anisotropic spatial fractional Riesz derivatives. Fully discrete numerical methods are proposed using exponential Strang's splitting schemes for the time integration with finite difference discretization in space. It is proved that the proposed methods can preserve the discrete maximum principle unconditionally. Furthermore, the fully discrete methods are theoretically confirmed to be convergent with second-order accuracy in both of time and space. In practical implementation, the proposed algorithms require to compute the matrix exponential associated with only one-dimensional discretized matrices that possess Toeplitz structure. Meanwhile, a fast algorithm is further developed for evaluating the product of the Toeplitz matrix exponential with a vector. Numerical examples are presented to verify the theoretical analysis and demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

For their several attractive features from the viewpoint of the numerical computations, linear barycentric rational interpolants have been recently used to construct various numerical methods for solving different classes of equations. In this paper, we introduce a family of linear multistep second derivative methods together with a starting procedure based on barycentric rational interpolants. The order of convergence and linear stability properties of the proposed methods have been investigated. To validate the theoretical results and efficiency of the methods, some numerical experiments have been provided. [ABSTRACT FROM AUTHOR]

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge–Kutta methods and continuous-stage Runge–Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters, a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators. [ABSTRACT FROM AUTHOR]

Sixty years ago, Butcher (Butcher Math. Soc. 3, 185–201 1963) characterized a natural tabulation of the order conditions for Runge–Kutta methods of order p as an isomorphism from the set of rooted trees having up to p nodes and provided examples of explicit and implicit methods of several orders. Within a few years, Fehlberg (Fehlberg 1968) derived pairs of explicit methods of successive orders that could be implemented efficiently by using the difference of each pair of estimates to control the local error. Unfortunately, Fehlberg's pairs were deficient for quadrature problems. Subsequently, this author (Verner SIAM J. Numer. Anal. 15, 772–790 1978, Verner SIAM J. Numer. Anal. 16, 857–875 1979) derived parametric families of explicit Runge–Kutta pairs of increasing orders 6 to 9 that avoided this problem altogether. These, and most known explicit methods, have been derived by exploiting certain "simplifying conditions" suggested by Butcher (Butcher Math. Soc. 3, 185–201 1963, Butcher 1987 [pp. 194 f.]) that imposed constraints on subsets of the coefficients and thereby simplified the solution of the order conditions for moderate to high-order methods. "RK-Test-21", a MAPLE program developed recently by Butcher (Butcher 2021), was applied to derive known 13-stage pairs of orders 7 and 8. Unexpectedly, results of this application revealed the existence of some previously unknown pairs—i.e., some with the correct orders that satisfied most, but not all, of the previously known simplifying conditions. This present study develops formulas for directly computing exact coefficients of these new pairs together with others lying within this new parametric family of (13,7-8) pairs. While the best of these new pairs falls short of the best of pairs already known, the properties discovered might be utilized to precisely characterize recently reported higher-order methods found using other approaches by Khashin (Khashin 2009) and Zhang (Zhang 2019) and possibly lead to finding other Runge–Kutta and related yet unknown methods. [ABSTRACT FROM AUTHOR]

We present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

This article considers the numerical treatment of piecewise-smooth dynamical systems. Classical solutions as well as sliding modes up to codimension-2 are treated. An algorithm is presented that, in the case of non-uniqueness, selects a solution that is the formal limit solution of a regularized problem. The numerical solution of a regularized differential equation, which creates stiffness and often also high oscillations, is avoided. [ABSTRACT FROM AUTHOR]