scholarly journals Modelling and numerical approximation for the nonconservative bitemperature Euler model

2018 ◽  
Vol 52 (4) ◽  
pp. 1353-1383 ◽  
Author(s):  
D. Aregba–Driollet ◽  
J. Breil ◽  
S. Brull ◽  
B. Dubroca ◽  
E. Estibals
Keyword(s):  
Kinetic Model ◽  
Numerical Approximation ◽  
Hydrodynamic Limit ◽  
Euler System ◽  
Fluid System ◽  
The Poisson Equation ◽  
The Family ◽  
Euler Model ◽  
Dissipation Property ◽  
Linear Relaxation Approach

This paper is devoted to the study of the nonconservative bitemperature Euler system. We firstly introduce an underlying two species kinetic model coupled with the Poisson equation. The bitemperature Euler system is then established from this kinetic model according to an hydrodynamic limit. A dissipative entropy is proved to exist and a solution is defined to be admissible if it satisfies the related dissipation property. Next, four different numerical methods are presented. Firstly, the kinetic model gives rise to kinetic schemes for the fluid system. The second approach belongs to the family of the discrete BGK schemes introduced by Aregba–Driollet and Natalini. Finally, a quasi-linear relaxation approach and a Lagrange-remap scheme are considered.

scholarly journals Nonconservative hyperbolic systems in fluid mechanics

2018 ◽  
Vol 64 ◽  
pp. 1-16
Author(s):  
Denise Aregba-Driollet ◽  
Stéphane Brull ◽  
Xavier Lhébrard
Keyword(s):  
Kinetic Model ◽  
Magnetic Fields ◽  
Fluid Mechanics ◽  
Numerical Approximation ◽  
Hyperbolic Systems ◽  
Euler System ◽  
Kinetic Approach ◽  
Nonconservative Hyperbolic Systems

This paper is devoted to the numerical approximation of nonconservative hyperbolic systems. More precisely, we consider the bitemperature Euler system and we propose two methods of discretization. The first one is a kinetic approach based on an underlying kinetic model. The second one deals with a Suliciu approach when magnetic fields are taken into account.

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

2019 ◽  
Vol 25 (1) ◽  
pp. 37-60
Author(s):  
Antoon Pelsser ◽  
Kossi Gnameho
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Mathematical Finance ◽  
Backward Stochastic Differential Equations ◽  
Terminal Condition ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
The Family ◽  
Infinite Dimensional System

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

scholarly journals New conformal mapping for adaptive resolving of the complex singularities of Stokes wave

2017 ◽  
Vol 473 (2202) ◽  
pp. 20170198 ◽  
Author(s):  
Pavel M. Lushnikov ◽  
Sergey A. Dyachenko ◽  
Denis A. Silantyev
Keyword(s):  
Free Surface ◽  
Conformal Mapping ◽  
Numerical Approximation ◽  
Travelling Wave ◽  
Stokes Wave ◽  
Numerical Convergence ◽  
Complex Singularities ◽  
The Family ◽  
Stokes Waves ◽  
Numerical Grid

A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.

scholarly journals Asymptotic analysis of Vlasov-type equations under strong local alignment regime

2015 ◽  
Vol 25 (11) ◽  
pp. 2153-2173 ◽  
Author(s):  
Moon-Jin Kang ◽  
Alexis F. Vasseur
Keyword(s):  
Kinetic Equation ◽  
Hyperbolic Conservation Laws ◽  
Hydrodynamic Limit ◽  
Local Equilibrium ◽  
Euler System ◽  
Entropy Method ◽  
Local Alignment ◽  
Hyperbolic Conservation ◽  
Self Organized ◽  
Relative Entropy Method

We consider the hydrodynamic limit of a collisionless and non-diffusive kinetic equation under strong local alignment regime. The local alignment is first considered by Karper, Mellet and Trivisa in [On strong local alignment in the kinetic Cucker–Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics & Statistics, Vol. 49 (Springer, 2014), pp. 227–242], as a singular limit of an alignment force proposed by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 141 (2011) 923–947]. As the local alignment strongly dominates, a weak solution to the kinetic equation under consideration converges to the local equilibrium, which has the form of mono-kinetic distribution. We use the relative entropy method and weak compactness to rigorously justify the weak convergence of our kinetic equation to the pressureless Euler system.

scholarly journals On parameter estimation with the Wasserstein distance

2019 ◽  
Vol 8 (4) ◽  
pp. 657-676 ◽  
Author(s):  
Espen Bernton ◽  
Pierre E Jacob ◽  
Mathieu Gerber ◽  
Christian P Robert
Keyword(s):  
Numerical Approximation ◽  
Empirical Distribution ◽  
Asymptotic Properties ◽  
Wasserstein Distance ◽  
Generative Models ◽  
Misspecified Models ◽  
Likelihood Functions ◽  
The Family ◽  
Data Generating Process ◽  
Approximate Bayesian

Abstract Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptotic properties of such minimum Wasserstein distance estimators, complementing results derived by Bassetti, Bodini and Regazzini in 2006. In particular, our results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model. Our results are motivated by recent applications of minimum Wasserstein estimators to complex generative models. We discuss some difficulties arising in the numerical approximation of these estimators. Two of our numerical examples ($g$-and-$\kappa$ and sum of log-normals) are taken from the literature on approximate Bayesian computation and have likelihood functions that are not analytically tractable. Two other examples involve misspecified models.

scholarly journals Kinetic Model for pH Variation Resulted from Interaction of Aqueous Solution Contaminated with Nickel Ions and Cement Kiln Dust

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Dooraid N. Ahmed ◽  
Ayad A. H. Faisal ◽  
Salim H. Jassam ◽  
Laith A. Naji ◽  
Mu. Naushad
Keyword(s):  
Kinetic Model ◽  
Adsorption Capacity ◽  
Numerical Approximation ◽  
Cement Kiln Dust ◽  
Cement Kiln ◽  
Solution Ph ◽  
Nickel Ions ◽  
Ph Variation ◽  
Reaction Constant ◽  
Kiln Dust

Reaction term in the transport equation which described the migration of metal ions in the porous medium is frequently represented by conventional kinetic models such as pseudo-first order, pseudo-second order, and others. Unfortunately, these models are applicable for the constant value of solution pH, and they cannot simulate the real situation in the field scale where this pH may be changed with time. Accordingly, the present study is a good attempt to derive the kinetic model that can simulate the change in the pH of the solution through solute transport. This was achieved by modifying the adsorption capacity and reaction constant to be functions in terms of solution pH by using semianalytical analysis and numerical approximation. The results proved that the kinetic model based on the numerical approximation (using exponential functions for adsorption capacity and reaction constant) symbolled as model 2 was more representative from other models applied for the description of interaction of nickel ions (with initial concentration of 400 mg/L) and cement kiln dust with sum of squared error ≤1.54913 and determination coefficient ≥0.889. Also, the developed models had high ability for recognizing between pure precipitation and pure adsorption.

scholarly journals Numerical approximation to the general kinetic model for ASL quantification

2020 ◽  
Vol 84 (5) ◽  
pp. 2846-2857
Author(s):  
Nam G. Lee ◽  
Ahsan Javed ◽  
Terrence R. Jao ◽  
Krishna S. Nayak
Keyword(s):  
Kinetic Model ◽  
Numerical Approximation

scholarly journals Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jeongho Kim ◽  
Bora Moon
Keyword(s):  
Quantum Correction ◽  
Hydrodynamic Limit ◽  
Energy Estimate ◽  
Correction Term ◽  
Euler System ◽  
Entropy Method ◽  
Planck Constant ◽  
Nonlinear Schrödinger ◽  
Relative Entropy Method ◽  
Chern Simons

<p style='text-indent:20px;'>We present two types of the hydrodynamic limit of the nonlinear Schrödinger-Chern-Simons (SCS) system. We consider two different scalings of the SCS system and show that each SCS system asymptotically converges towards the compressible and incompressible Euler system, coupled with the Chern-Simons equations and Poisson equation respectively, as the scaled Planck constant converges to 0. Our method is based on the modulated energy estimate. In the case of compressible limit, we observe that the classical theory of relative entropy method can be applied to show the hydrodynamic limit, with the additional quantum correction term. On the other hand, for the incompressible limit, we directly estimate the modulated energy to derive the desired asymptotic convergence.</p>

Fully nonlinear periodic internal waves in a two-fluid system of finite depth

2010 ◽  
Vol 652 ◽  
pp. 259-298 ◽  
Author(s):  
R. CAMASSA ◽  
P.-O. RUSÅS ◽  
A. SAXENA ◽  
R. TIRON
Keyword(s):  
Closed Form ◽  
Internal Waves ◽  
Solitary Waves ◽  
Travelling Wave ◽  
Euler System ◽  
Layer System ◽  
Fluid System ◽  
Model Solutions ◽  
Existence Domain ◽  
Two Fluid

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.

scholarly journals Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

2021 ◽  
pp. 1-83
Author(s):  
Young-Pil Choi ◽  
Jinwook Jung
Keyword(s):  
Boundary Condition ◽  
Asymptotic Analysis ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Stokes System ◽  
Dirichlet Boundary ◽  
Euler System ◽  
Navier Stokes ◽  
Fluid System ◽  
Fokker Planck

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

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