Numerical approximations of one-dimensional linear conservation
equations with discontinuous coefficients

Gosse, Laurent; James, François

doi:10.1090/s0025-5718-00-01185-6

Cited by 61 publications

(66 citation statements)

References 24 publications

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“…Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in [5,9,12,24,32,36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17,19,20,22,23]; see also [8,21,40,39] for closely related works. We stress that our results do not contradict [26].…”

Section: Introductionmentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

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Section: Introductionmentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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“…In this direction, among others, [4,6,27,36] address the issue of linearizing the system around solutions developing shock discontinuities. There has been also an extensive research regarding the corresponding adjoint system and its discretization (see, for instance, [5,23,28]). In [19,20] the authors analyze the pointwise convergence of the linearized and adjoint approximations for discontinuous solutions in a discretize-then-optimize approach.…”

Section: Introductionmentioning

confidence: 99%

Numerical aspects of large-time optimal control of Burgers equation

2016

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Abstract.In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical approximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since numerical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax−Friedrichs scheme (MLF) and the Engquist−Osher (EO) one. It is by now wellknown that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT).Mathematics Subject Classification. 49M, 35Q35, 65M06.

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“…In this paper, we are interested in the scalar conservation law with the linear flux function involving discontinuous coefficients as follows [14,20]:…”

Section: Introductionmentioning

confidence: 99%

“…The Dirac measure-valued solution was introduced in the Riemann solutions in some non-classical situations and then he discovered that the Dirac measure-valued solution can be obtained as the limit of the following self-similar viscosity regularized system (1.4) k t = εtk xx , u t + (ku) x = εtu xx , by using the standard Dafermos technique [7,8]. In [14], Gosse and James have also obtained the Dirac measure-valued solution to the Riemann problem (1.2) and (1.3) in the numerical computation.…”

Section: Introductionmentioning

confidence: 99%

The Delta Standing Wave Solution for the Linear Scalar Conservation Law With Discontinuous Coefficients Using a Self-Similar Viscous Regularization

Li¹,

Shen²

2015

Bulletin of the Korean Mathematical Society

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Abstract. This paper is mainly concerned with the formation of delta standing wave for the scalar conservation law with a linear flux function involving discontinuous coefficients by using the self-similar viscosity vanishing method. More precisely, we use the self-similar viscosity to smooth out the discontinuous coefficient such that the existence of approximate viscous solutions to the delta standing wave for the Riemann problem is established and then the convergence to the delta standing wave solution is also obtained when the viscosity parameter tends to zero. In addition, the Riemann problem is also solved with the standard method and the instability of Riemann solutions with respect to the specific small perturbation of initial data is pointed out in some particular situations.

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Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients

Cited by 61 publications

References 24 publications

Localization effects and measure source terms in numerical schemes for balance laws

Localization effects and measure source terms in numerical schemes for balance laws

Numerical aspects of large-time optimal control of Burgers equation

The Delta Standing Wave Solution for the Linear Scalar Conservation Law With Discontinuous Coefficients Using a Self-Similar Viscous Regularization

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