Related papers: High order linearly implicit methods for semilinea…
A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach…
High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade…
Problems that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, where…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary…
In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with…
Low-storage explicit Runge-Kutta schemes are particularly popular for the numerical integration of time-dependent partial differential equations based on the method-of-lines due to their efficiency and their reduced memory requirements. We…
A new preconditioner based on a block $LDU$ factorization with algebraic multigrid subsolves for scalability is introduced for the large, structured systems appearing in implicit Runge-Kutta time integration of parabolic partial…
This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned…
In this work, we present approaches to rigorously certify $A$- and $A(\alpha)$-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is…
In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge--Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general…
Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case…
In this paper we consider an approach to improve the performance of exponential Runge--Kutta integrators and Lawson schemes} in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for…
We develop continuous-stage Runge-Kutta methods based on weighted orthogonal polynomials in this paper. There are two main highlighted merits for developing such methods: Firstly, we do not need to study the tedious solution of…
This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods…
The existing discrete variational derivative method is only second-order accurate and fully implicit. In this paper, we propose a framework to construct an arbitrary high-order implicit (original) energy stable scheme and a second-order…
In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the the real direction with the square of polynomial…
We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order…
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…