Related papers: Numerical simulation of nonlinear continuity equat…
The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational…
We consider numerical methods for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients defined by a one parameter family of interface conditions at the discontinuity. We construct immersed…
We prove the existence of nonnegative weak solutions to a class of second and fourth order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space $\mathbb{R}^d$, for…
Numerical schemes that conserve invariants have demonstrated superior performance in various contexts, and several unified methods have been developed for constructing such schemes. However, the mathematical properties of these schemes…
Accurate and efficient prediction of multi-scale flows remains a formidable challenge. Constructing theoretical models and numerical methods often involves the design and optimization of parameters. While gradient descent methods have been…
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to…
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an…
In this work, we obtain the numerical temperature field to a thermally developing fluid flow inside parallel plates problem with a quantum computing method. The physical problem deals with the heat transfer of a steady state,…
The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends…
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation of the form $\partial_t u = \partial_x(u \cdot c[\partial_x(h^\prime(u)+v)])$ on an interval. This scheme will consist of a spatio-temporal…
This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model…
We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.
This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with…
We present and analyze three distinct semi-discrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method, and the finite element…
In this paper, we consider numerical approximation of an electrically conductive ferrofluid model, which consists of Navier-Stokes equations, magnetization equation, and magnetic induction equation. To solve this highly coupled, nonlinear,…
Non-Newtonian fluid flow might be driven by spatially nonlocal velocity, the dynamics of which can be described by promising fractional derivative models. This short communication proposes a space FrActional-order Constitutive Equation…
We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability properties in the…
In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal,…
This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient…
We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly…