Quantity of the inverse problem data for the system of conservation laws

Abstract In this paper we study properties of the model, that describes the plane acoustic waves propagation. The model is based on the hyperboliv system of PDE, which is solved numerically by using the finite-volume method, based on Godunov scheme. After studying the direct problem we turn to the inverse one, where our goal is to recover the parameters of the system of PDE by using the initial data, measured in the receivers. We obtain the formula for the gradient of the misfits functional, which allows us to apply gradient-based optimization for recovering the density of the medium. We present the results of numerical experiments for different number of receivers, thus, studying the influence of the quantity of the data of inverse problem on the accuracy of the solution.

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A LATTICE BOLTZMANN METHOD-BASED FLUX SOLVER AND ITS APPLICATION TO SOLVE SHOCK TUBE PROBLEM

Modern Physics Letters B ◽

10.1142/s021798490901828x ◽

2009 ◽

Vol 23 (03) ◽

pp. 313-316 ◽

Cited By ~ 18

Author(s):

C. Z. JI ◽

C. SHU ◽

N. ZHAO

Keyword(s):

Lattice Boltzmann Method ◽

Lattice Boltzmann ◽

Numerical Experiments ◽

Compressible Flows ◽

Navier Stokes ◽

Godunov Scheme ◽

Lattice Boltzmann Models ◽

Volume Method ◽

Boltzmann Method ◽

Shock Profile

This paper presents an approach, which combines the conventional finite volume method (FVM) with the lattice Boltzmann Method (LBM), to simulate compressible flows. Similar to the Godunov scheme, in the present approach, LBM is used to evaluate the flux at the interface for local Riemann problem when solving Euler/Navier-Stokes (N-S) equations by FVM. Two kinds of popular compressible Lattice Boltzmann models are applied in the new scheme, and some numerical experiments are performed to validate the proposed approach. From the sharper shock profile and higher computational efficiency, numerical results demonstrate that the proposed scheme is superior to the conventional Godunov scheme. It is expected that the proposed scheme has a potential to become an efficient flux solver in solving compressible Euler/N-S equations.

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Numerics of acoustical 2D tomography based on the conservation laws

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0061 ◽

2020 ◽

Vol 28 (2) ◽

pp. 287-297 ◽

Cited By ~ 2

Author(s):

Sergey I. Kabanikhin ◽

Dmitriy V. Klyuchinskiy ◽

Nikita S. Novikov ◽

Maxim A. Shishlenin

Keyword(s):

Mathematical Modeling ◽

Inverse Problem ◽

Conservation Laws ◽

Speed Of Sound ◽

Gradient Method ◽

Acoustic Waves ◽

Order System ◽

First Order ◽

Waves Propagation ◽

Gradient Of The Functional

AbstractWe investigate the mathematical modeling of the 2D acoustic waves propagation, based on the conservation laws. The hyperbolic first-order system of partial differential equations is considered and solved by the method of S. K. Godunov. The inverse problem of reconstructing the density and the speed of sound of the medium is considered. We apply the gradient method to reconstruct the parameters of the medium. The gradient of the functional is obtained. Numerical results are presented.

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Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations

Mathematics ◽

10.3390/math9020199 ◽

2021 ◽

Vol 9 (2) ◽

pp. 199

Author(s):

Dmitriy Klyuchinskiy ◽

Nikita Novikov ◽

Maxim Shishlenin

Keyword(s):

Inverse Problem ◽

Finite Number ◽

Hyperbolic System ◽

Speed Of Sound ◽

Acoustic Waves ◽

Coefficient Inverse Problem ◽

Optimization Scheme ◽

First Order ◽

Acoustic Equations ◽

Gradient Based

We consider the coefficient inverse problem for the first-order hyperbolic system, which describes the propagation of the 2D acoustic waves in a heterogeneous medium. We recover both the denstity of the medium and the speed of sound by using a finite number of data measurements. We use the second-order MUSCL-Hancock scheme to solve the direct and adjoint problems, and apply optimization scheme to the coefficient inverse problem. The obtained functional is minimized by using the gradient-based approach. We consider different variations of the method in order to obtain the better accuracy and stability of the appoach and present the results of numerical experiments.

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The Method of Computing the Thermal Conductivity of Heat-Insulation Oil Pipe Coupling

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.278-280.251 ◽

2013 ◽

Vol 278-280 ◽

pp. 251-255

Author(s):

Li Jun Liu ◽

Peng Yu ◽

Ying Xu

Keyword(s):

Thermal Conductivity ◽

Inverse Problem ◽

Heat Conduction ◽

Temperature Distribution ◽

Objective Function ◽

Direct Problem ◽

Heat Insulation ◽

Pipe Coupling ◽

Volume Method ◽

Insulation Oil

Based on the inverse problem theory of heat conduction, thermal conductivity of heat-insulation oil pipe coupling is researched in the paper. The finite volume method is extended to solve the direct problem, and the 0.618 method is used to solve the inverse problem by optimizing the objective function. The results show that by using the method, taking the outer wall’s temperature as the sample indicators, the thermal conductivity and the temperature distribution of heat-insulation oil pipe coupling and the temperature distribution of heat-insulation oil pipe can be obtained easily and accurately.

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Some features of solving an inverse backward problem for a generalized Burgers’ equation

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0078 ◽

2020 ◽

Vol 28 (5) ◽

pp. 641-649

Author(s):

Dmitry V. Lukyanenko ◽

Igor V. Prigorniy ◽

Maxim A. Shishlenin

Keyword(s):

Inverse Problem ◽

Direct Problem ◽

Numerical Experiments ◽

Burgers Equation ◽

Simple Algorithm ◽

Singularly Perturbed ◽

Generalized Burgers Equation ◽

Backward Problem ◽

Singularly Perturbed Parabolic Equation ◽

The Cost

AbstractIn this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.

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A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry074 ◽

2018 ◽

Vol 40 (1) ◽

pp. 405-421 ◽

Cited By ~ 1

Author(s):

N Chatterjee ◽

U S Fjordholm

Keyword(s):

Initial Data ◽

Conservation Laws ◽

Finite Volume Method ◽

Finite Volume ◽

Numerical Examples ◽

Continuity Equations ◽

Multiple Dimensions ◽

Nonlocal Conservation Laws ◽

Volume Method ◽

Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽

10.3390/math9040310 ◽

2021 ◽

Vol 9 (4) ◽

pp. 310 ◽

Cited By ~ 1

Author(s):

Pedro Ortiz ◽

Juan Carlos Trillo

Keyword(s):

Initial Data ◽

Numerical Experiments ◽

Approximation Order ◽

Piecewise Polynomial ◽

Nonuniform Grids ◽

Uniform Grids ◽

Reconstruction Operator ◽

Convexity Properties ◽

Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

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