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This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A…
In this work, we study and extend a class of semi-Lagrangian exponential methods, which combine exponential time integration techniques, suitable for integrating stiff linear terms, with a semi-Lagrangian treatment of nonlinear advection…
A formulation for the efficient calculation of the electromagnetic retarded potential generated by time-dependent electron density in the context of real-time time dependent density functional theory (RT-TDDFT) is presented. The electron…
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the…
In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…
We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure…
We report in this paper the analysis for the linear and nonlinear version of the flux corrected transport (FEM-FCT) scheme in combination with the backward Euler time-stepping scheme applied to time-dependent convection-diffusion-reaction…
In this article, we employ the construction of the time-marching Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for non-linear systems of ordinary differential…
In this work, we study time-splitting strategies for the numerical approximation of evolutionary reaction-diffusion problems. In particular, we formulate a family of domain decomposition splitting methods that overcomes some typical…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators…
Sequential-in-time methods solve a sequence of training problems to fit nonlinear parametrizations such as neural networks to approximate solution trajectories of partial differential equations over time. This work shows that…
Extrapolation remains a grand challenge in deep neural networks across all application domains. We propose an operator learning method to solve time-dependent partial differential equations (PDEs) continuously and with extrapolation in time…
Some properties of a Local discontinuous Galerkin (LDG) algorithm are demonstrated for the problem of evaluting a second derivative $g = f_{xx}$ for a given $f$. (This is a somewhat unusual problem, but it is useful for understanding the…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are controlled by the interaction with the slow system. Using…
We introduce a dimensional splitting method based on the intertwining property of the Radon transform, with a particular focus on its applications related to hyperbolic partial differential equations (PDEs). This dimensional splitting has…
The nonlinear coupled reaction-diffusion (NCRD) systems are important in the formation of spatiotemporal patterns in many scientific and engineering fields, including physical and chemical processes, biology, electrochemical processes,…
In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and…
Exploration in sparse reward environments remains a significant challenge in reinforcement learning, particularly in Contextual Markov Decision Processes (CMDPs), where environments differ across episodes. Existing episodic intrinsic…