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The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…
By means of a systematic expansion around the infinite-dimensional solution, we obtain an approximation scheme to compute properties of glasses in low dimensions. The resulting equations take as input the thermodynamic and structural…
We present a numerical scheme for approximating the incompressible Navier-Stokes equations based on an auxiliary variable associated with the total system energy. By introducing a dynamic equation for the auxiliary variable and…
The problems of numerical modeling of viscous incompressible fluid flows are widely considered in computational fluid dynamics. Stationary solutions of boundary value problems for the Navier-Stokes equations exist at large Reynolds numbers,…
We present effective numerical algorithms for locally recovering unknown governing differential equations from measurement data. We employ a set of standard basis functions, e.g., polynomials, to approximate the governing equation with high…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…
Results about existence and uniqueness of solutions of initial value problem for certain types of partial differential equations are recalled as well as iterative scheme and an error estimate for approximate solutions obtained using this…
This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in…
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\big\|u(t,\cdot)-u^\ve(t,\cdot)\big\|_{\L^1}= \O(1)(1+t)\cdot \sqrt\ve|\ln\ve|$ on the distance between an exact BV solution $u$ and…
The system of equations of one-dimensional shallow water over uneven bottom in Euler's and Lagrange's variables is considered. Intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of…
The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity…
We propose an implementable, neural network-based structure preserving probabilistic numerical approximation for a generalized obstacle problem describing the value of a zero-sum differential game of optimal stopping with asymmetric…
In this paper, we introduce a hyperbolic model for entropy dissipative system of viscous conservation laws via a flux relaxation approach. We develop numerical schemes for the resulting hyperbolic relaxation system by employing the…
We consider coupled models for particulate flows, where the disperse phase is made of particles with distinct sizes. We are thus led to a system coupling the incompressible Navier-Stokes equations to the multi-component Vlasov-Fokker-Planck…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via…
We consider initial value problems of nonlinear dynamical systems, which include physical parameters. A quantity of interest depending on the solution is observed. A discretisation yields the trajectories of the quantity of interest in many…
Motivated by the statistical description of turbulence, we study statistical conservation laws in the form of kinetic-type PDEs for joint probability density functions (PDFs) and cumulative distribution functions (CDFs) associated with…
When fully conservative methods are used to simulate transcritical flow, spurious pressure oscillations and numerical instability are generated. The strength and speed of propagation of shock waves cannot be represented correctly using a…
Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article,…