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We study multiscale integrator numerical schemes for a class of stiff stochastic differential equations (SDEs). We consider multiscale SDEs with potentially multiple attractors that behave as diffusions on graphs as the stiffness parameter…
We present an explicit method for simulating stochastic differential equations (SDEs) that have variable diffusion coefficients and satisfy the detailed balance condition with respect to a known equilibrium density. In Tupper and Yang…
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary…
We present an explicit multiscale algorithm for solving differential equations for problems with high-frequency modes that can be averaged over by separating and scaling the fast and slow dynamics within a single equation. We introduce a…
In this paper, we implement exponential integrators, specifically Integrating Factor (IF) and Exponential Time Differencing (ETD) methods, using pseudo-spectral techniques to solve phase-field equations within a Python framework. These…
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence…
The aim of this paper is to introduce several new particle representations for \textit{ergodic} McKean-Vlasov SDEs. We construct new algorithms by leveraging recent progress in weak convergence analysis of interacting particle system. We…
We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based…
Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable…
This work systematically investigates the performance of FORCE--$\alpha$ numerical fluxes within an arbitrary high order semidiscrete finite volume (FV) framework for hyperbolic partial differential equations (PDEs). Such numerical fluxes…
A component-splitting method is proposed to improve convergence characteristics for implicit time integration of compressible multicomponent reactive flows. The characteristic decomposition of flux jacobian of multicomponent Navier-Stokes…
Lagrangian averaging plays an important role in the analysis of wave--mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is however challenging. Typical…
There has been an increasing interest in developing efficient immersed boundary method (IBM) based on Cartesian grids, recently in the context of high-order methods. IBM based on volume penalization is a robust and easy to implement method…
Stiff systems of ordinary differential equations (ODEs) and sparse training data are common in scientific problems. This paper describes efficient, implicit, vectorized methods for integrating stiff systems of ordinary differential…
In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in…
In this work, high-order discrete well-balanced methods for one-dimensional hyperbolic systems of balance laws are proposed. We aim to construct a method whose discrete steady states correspond to solutions of arbitrary high-order ODE…
Solving partial differential equations is difficult. Recently proposed neural resolution-invariant models, despite their effectiveness and efficiency, usually require equispaced spatial points of data. However, sampling in spatial domain is…
Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall…
We introduce exponential numerical integration methods for stiff stochastic dynamical systems of the form $d\mathbf{z}_t = L(t)\mathbf{z}_tdt + \mathbf{f}(t)dt + Q(t)d\mathbf{W}_t$. We consider the setting of time-varying operators $L(t),…
We use a continuous mesoscopic model to address the yielding properties of plastic composites, formed by a host material and inclusions with different elastic and/or plastic properties. We investigate the flow properties of the composed…