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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…
Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to…
Computer simulations in QCD are based on the discretization of the theory on a Euclidean lattice. To compute the mean value of an observable, usually the Hybrid Monte Carlo method is applied. Here equations of motion, derived from an…
High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The…
The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong…
We study the convergence of a class of Runge-Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability…
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…
Cloud and precipitation microphysics packages in atmospheric general circulation models typically use first-order time integration methods with a large time step, requiring ad hoc limiters and substepping of the sedimentation scheme to…
Integration by parts reduction is a standard component of most modern multi-loop calculations in quantum field theory. We present a novel strategy constructed to overcome the limitations of currently available reduction programs based on…
We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid…
Many important initial value problems have the property that energy is non-increasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for…
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…
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…
Isospectral Runge-Kutta methods are well-suited for the numerical solution of isospectral systems such as the rigid body and the Toda lattice. More recently, these integrators have been applied to geophysical fluid models, where their…
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 present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order…
The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods.…
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 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…
We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the…