Related papers: R-adaptive multisymplectic and variational integra…
The advantage of particle Lagrangian methods in computational fluid dynamics is that advection is accurately modeled. However, this complicates the calculation of space derivatives. If a mesh is employed, it must be updated at each time…
This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector…
Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation…
Symplectic integration algorithms have become popular in recent years in long-term orbital integrations because these algorithms enforce certain conservation laws that are intrinsic to Hamiltonian systems. For problems with large variations…
Modern N-body techniques for planetary dynamics are generally based on symplectic algorithms specially adapted to the Kepler problem. These methods have proven very useful in studying planet formation, but typically require the timestep for…
The relativistic hydrodynamics (RHD) equations can give rise to solutions which have shocks, contact discontinuities, and other sharp structures, which interact and evolve over time. Capturing these sharp waves effectively requires a mesh…
In this paper, we introduce a new approach to solving the porous medium equation using a moving mesh finite element method that leverages the Onsager variational principle as an approximation tool. Both the continuous and discrete problems…
Simulation of contact and friction dynamics is an important basis for control- and learning-based algorithms. However, the numerical difficulties of contact interactions pose a challenge for robust and efficient simulators. A…
We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to…
This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic…
We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation…
We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, rather than the…
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
It is found that the wave functions of the Gross-Pitaevskii equation (GPE) often vary significantly in different spatial regions, with some components exhibiting sharp variations while others remain smooth. Solving the GPE on a single mesh,…
3D reconstruction and simulation, although interrelated, have distinct objectives: reconstruction requires a flexible 3D representation that can adapt to diverse scenes, while simulation needs a structured representation to model motion…
We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, SPH, and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both smoothed-particle…
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes…
We present a novel shape-approximating anisotropic re-meshing algorithm as a geometric generalization of the adaptive moving mesh method. Conventional moving mesh methods reduce the interpolation error of a mesh that discretizes a given…
The paper outlines novel variational technique for finding microstructures of optimal multimaterial composites, bounds of composites properties, and multimaterial optimal designs. The translation method that is used for the exact…
We present a real-space adaptive-coordinate method, which combines the advantages of the finite-difference approach with the accuracy and flexibility of the adaptive coordinate method. The discretized Kohn-Sham equations are written in…