Related papers: Geometry of Numerical Complex Time Integration
Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta…
Most numerical methods for time integration use real-valued time steps. Complex time steps, however, can provide an additional degree of freedom, as we can select the magnitude of the time step in both the real and imaginary directions. We…
In this paper, we present error estimates of fully discrete Runge--Kutta discontinuous Galerkin (DG) schemes for linear time-dependent partial differential equations. The analysis applies to explicit Runge--Kutta time discretizations of any…
We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
Complex dynamical networks appear in a wide range of physical, biological, and engineering systems. The coupling of subsystems with varying time scales often results in multirate behavior. During the simulation of highly integrated…
We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The…
Delayed processes are ubiquitous throughout biology. These delays may arise through maturation processes or as the result of complex multi-step networks, and mathematical models with distributed delays are increasingly used to capture the…
Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for…
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes…
The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\epsilon$, by solving only $$O\Big(\log N \log \frac1\epsilon \Big) $$ linear…
In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we…
A novel class of Runge-Kutta discontinuous Galerkin schemes for coupled systems of conservation laws in multiple space dimensions that are separated by a fixed sharp interface is introduced. The schemes are derived from a relaxation…
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…
A convolutional neural network can be constructed using numerical methods for solving dynamical systems, since the forward pass of the network can be regarded as a trajectory of a dynamical system. However, existing models based on…
Mixed-precision methods combine low and high precision arithmetics to exploit low precision computational speed and high precision accuracy. Large ODE systems that contain many heterogeneous interactions lead to a high computational cost…
The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method offers a…
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…
This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge-Kutta collocation formalism with…
A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically,…