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In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic mutli-symplectic…
We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line,…
We consider numerical simulation of the isotropic elastic wave equations arising from seismic applications with non-trivial land topography. The more flexible finite element method is applied to the shallow region of the simulation domain…
The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the…
An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived, by approximating the Hamiltonian formulation, based on the Poisson brackets and the vorticity-divergence variables, of the inviscid shallow water…
A new energy-consistent discretization of the viscous dissipation function in incompressible flows is proposed. It is implied by choosing a discretization of the diffusive terms and a discretization of the local kinetic energy equation and…
`Dual composition', a new method of constructing energy-preserving discretizations of conservative PDEs, is introduced. It extends the summation-by-parts approach to arbitrary differential operators and conserved quantities. Links to…
This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the…
Positive semi-definite expressions for the time-domain macroscopic energy density in passive, spatially nondispersive, dipolar continua are derived from the underlying microscopic Maxwell equations satisfied by classical models of discrete…
This work is devoted to the derivation and the matematical study of a new model for water-soluble polymers and metal ions interactions, which are used in chemistry for their wide range of applications. First, we motivate and derive a model…
In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized…
A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities…
Certain energy-conservative Galerkin discretizations for nonlinear dispersive wave equations have revealed an unusual convergence behavior: optimal convergence is attained when continuous Lagrange finite element spaces of odd polynomial…
We introduce a structure preserving discretization of stochastic rotating shallow water equations, stabilized with an energy conserving Casimir (i.e. potential enstrophy) dissipation. A stabilization of a stochastic scheme is usually…
Conservation of current and conservation of charge are nearly the same thing: when enough is known about charge movement, conservation of current can be derived from conservation of charge, in ideal dielectrics, for example. Conservation of…
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial…
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving…
The modeling of complicated time-evolving physical dynamics from partial observations is a long-standing challenge. Particularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to…
The treatment of complex geometries in Computational Fluid Dynamics applications is a challenging endeavor, which immersed boundary and cut-cell techniques can significantly simplify by alleviating the meshing process required by…