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Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets are regions of the flow that minimally mix with the…
We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the…
This paper introduces a sharp-interface approach to simulating fluid-structure interaction involving flexible bodies described by general nonlinear material models and across a broad range of mass density ratios. This new flexible-body…
This work presents an arbitrary Lagrangian Eulerian (ALE) method for the compressible two-phase flow ejecta transporting model with the HLLC-2D Riemann solver. We focus on researching the precise equation to describe the interactions…
Currently existing energy-stable parametric finite element methods for surface diffusion flow and other flows are usually limited to first-order accuracy in time. Designing a high-order algorithm for geometric flows that can also be…
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled…
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional…
Lyapunov's indirect method is an attractive method for analyzing stability of non-linear systems since only the stability of the corresponding linearized system needs to be determined. Unfortunately, the proof for finite-dimensional systems…
A computational approach is introduced for the study of the rheological properties of complex fluids and soft materials. The approach allows for a consistent treatment of microstructure elastic mechanics, hydrodynamic coupling, thermal…
We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this…
The fluid flow transport and hydrodynamic problems often take the form of hyperbolic systems of conservation laws. In this work we will present a new scheme of finite volume methods for solving these evolution equations. It is a family of…
The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from Lehrenfeld &…
We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an…
We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that…
We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range…
Understanding and quantifying chaos from data remains challenging. We present a data-driven method for estimating the largest Lyapunov exponent (LLE) from one-dimensional chaotic time series using machine learning. A predictor is trained to…
In this work, we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties. First, we…
Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice…
Iterative gradient-based optimization algorithms are widely used to solve difficult or large-scale optimization problems. There are many algorithms to choose from, such as gradient descent and its accelerated variants such as Polyak's Heavy…
We present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid. The main idea is to incorporate Peskin's regularized delta function approach [Acta…