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Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial…
In this work, we explore various relevant aspects of the Smoothed Particle Hydrodynamics regarding Burger's equation. The stability, precision, and efficiency of the algorithm are investigated in terms of different implementations. In…
This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the…
One of the limitations of the Lattice Boltzmann Method in simulating inertial flows is the coupling of the discretization of space to the velocity discretization. It requires an increase of the size of computational lattices in order to…
Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler…
This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the…
In this paper we develop adaptive numerical schemes for certain nonlinear variational problems. The discretization of the variational problems is done by representing the solution as a suitable frame decomposition, i.e., a complete, stable,…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to…
The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow…
We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical…
The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel…
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their…
This work uses high-order discontinuous Galerkin discretization techniques as a generic, parameter-free, and reliable tool to accurately predict transitional and turbulent flows through medical devices. Flows through medical devices are…
The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference,…
Flow in fractured porous media represents a challenge for discretization methods due to the disparate scales and complex geometry. Herein we propose a new discretization, based on the mixed finite element method and mortar methods. Our…
The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work,…
Mathematical models for flow through porous media typically enjoy the so-called maximum principles, which place bounds on the pressure field. It is highly desirable to preserve these bounds on the pressure field in predictive numerical…
We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula,…
A relativistic generalization of the inviscid Burgers equation was proposed by LeFloch, Makhlof, and Okutmustur and then investigated on a Schwarzschild background. Here, we extend their analysis to a Friedmann-Lemaitre-Robertson-Walker…