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In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which…
Relaxation Runge-Kutta methods reproduce a fully discrete dissipation (or conservation) of entropy for entropy stable semi-discretizations of nonlinear conservation laws. In this paper, we derive the discrete adjoint of relaxation…
We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK…
We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of…
A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some…
We present second-order optimally stable Implicit-Explicit (IMEX) Runge-Kutta (RK) schemes with application to a modified set of shallow water equations that can be used to model the dynamics of lava flows. The schemes are optimally stable…
When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity…
A time discretization method is called strongly stable, if the norm of its numerical solution is nonincreasing. It is known that, even for linear semi-negative problems, many explicit Runge--Kutta (RK) methods fail to preserve this…
Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the…
This paper investigates the energy conservation properties of explicit Runge--Kutta (RK) time discretizations for autonomous skew-symmetric systems. For linear problems, we present a general framework for constructing RK methods in which…
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…
Traditional time discretization methods use a single timestep for the entire system of interest and can perform poorly when the dynamics of the system exhibits a wide range of time scales. Multirate infinitesimal step (MIS) methods (Knoth…
Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point…
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…
In this work we present a new class of Runge-Kutta (RK) methods for solving systems of hyperbolic equations with a particular structure, generalization of a wave-equation. The new methods are {\it partially implicit} in the sense that a…
Strong Stability Preserving (SSP) time integration schemes maintain stability of the forward Euler method for any initial value problem. However, only a small subset of Runge-Kutta (RK) methods are SSP, and many efficient high-order time…
Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux…
Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account…
Many time-dependent partial differential equations (PDEs) can be transformed into an ordinary differential equations (ODEs) containing moderately stiff and non-stiff terms after spatial semi-discretization. In the present paper, we…
We analyze the stability and accuracy (up to third order) of a new family of implicit-explicit Runge-Kutta (IMEX RK) methods. This analysis expedites development of methods with various balances in the number of explicit stages and implicit…