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Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized…
This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and…
We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we introduce a novel stability analysis, that covers the multi-rate extensions of all standard Runge-Kutta methods and allows to…
The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are…
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other…
A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the…
Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods.…
Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently,…
We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume…
A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive…
Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods…
Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…
We prove that Runge-Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments -- based on spectral analysis, resolvent condition or strong…
We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is…
In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating…
We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation…
Explicit Runge-Kutta (RK) integration of hyperbolic initial-boundary value problems with time-dependent Dirichlet data often displays order reduction: the observed convergence order falls below the nominal order because the stage structure…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The…
In this work, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov--Fokker--Planck system coupled with Poisson or Amp\`ere equation. Splitting methods…