Related papers: Stability-Optimized High Order Methods and Stiffne…
For turbulent problems of industrial scale, computational cost may become prohibitive due to the stability constraints associated with explicit time discretization of the underlying conservation laws. On the other hand, implicit methods…
Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In general, traditional explicit time integration schemes with restricted time step sizes are not…
Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been…
Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and…
This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a…
Stochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units,…
We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a…
We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability…
Dynamic state estimation (DSE) is becoming increasingly important for monitoring inverter-dominated power systems. Due to their cascading control structures, inverter-based resources (IBRs) exhibit multi-timescale dynamics, leading to stiff…
Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are…
The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to…
Neural PDE solvers offer a powerful tool for modeling complex dynamical systems, but often struggle with error accumulation over long time horizons and maintaining stability and physical consistency. We introduce a multiscale implicit…
The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travels at the…
In this article, we construct a numerical method for a stochastic version of the Susceptible Infected Susceptible (SIS) epidemic model, expressed by a suitable stochastic differential equation (SDE), by using the semi-discrete method to a…
We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable…
In this work, we introduce a self-adaptive implicit-explicit (IMEX) time integration scheme, named IMEX-RB, for the numerical integration of systems of ordinary differential equations (ODEs), arising from spatial discretizations of partial…
Stochastic differential equations (SDEs) are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to…
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and…
Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence…
Stiff dynamical systems represent a central challenge in multi scale modeling across combustion, chemical kinetics, and nonlinear dynamical systems. Neural operator learning has recently emerged as a promising approach to approximate…