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The nonlinear Schr\"odinger equation (NLSE) is a rich and versatile model, which in one spatial dimension has stationary solutions similar to those of the linear Schr\"odinger equation as well as more exotic solutions such as solitary waves…
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…
This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the…
In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent…
We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in…
An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve…
An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential…
The time-dependent one-dimensional nonlinear Schr\"odinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm that connects a pseudospectral step for the Hamiltonian term with a variational step…
The nonlinear Schr\"{o}dinger (NLS) equation possesses an infinite hierarchy of conserved densities and the numerical preservation of some of these quantities is critical for accurate long-time simulations, particularly for multi-soliton…
In this paper, we implement the Adaptive Moving Mesh method (AMM) to the solution of initial value problems involving the Schr\"odinger equation, and more specifically the Schr\"odinger-Poisson system of equations. This method is based on…
We propose a moving mesh adaptive approach for solving time-dependent partial differential equations. The motion of spatial grid points is governed by a moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a…
Astrophysical relativistic flow problems require high resolution three-dimensional numerical simulations. In this paper, we describe a new parallel three-dimensional code for simulations of special relativistic hydrodynamics (SRHD) using…
Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems…
The nonlinear Schr\"odinger equation (NLSE) is a fundamental model for wave dynamics in nonlinear media ranging from optical fibers to Bose-Einstein condensates. Accurately estimating its parameters, which are often strongly correlated,…
An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the…
In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics…
The Non-Markovian Stochastic Schrodinger Equation (NMSSE) offers a promising approach for open quantum simulations, especially in large systems, owing to its low scaling complexity and suitability for parallel computing. However, its…
Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have introduced this strategy before to…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some…