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On the numerical discretization of optimal control problems for conservation laws

Schäfer Aguilar Paloma, Schmitt Johann Michael, Ulbrich Stefan, Moos Michael

Control and Cybernetics

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2019

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Vol. 48, No. 2

345--376

EN

We analyze the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal control problems governed by entropy solutions of conservation laws. The difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and discontinuous end data. We derive the discrete adjoint scheme for monotone difference schemes in conservation form. It is known that convergence of the discrete adjoint can only be expected if the numerical scheme has viscosity of order O(h) with appropriate 0 < α < 1, which leads to quite viscous shock profiles. We show that by a slight modification of the end data of the discrete adjoint scheme, convergence to the correct reversible solution can be obtained also for numerical schemes with viscosity of order O(h) and with sharp shock resolution. The theoretical findings are confirmed by numerical results.

2

Solutions of incompressible Navier-Stokes equations with the artificial compressibility method for two- and three-dimensional shear-driven cavity flows

Kosma Z.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 1

113-125

EN

An efficient method for simulating laminar flows in complex geometries is presented. The artificial compressibility method was applied to solve two- and three-dimensional Navier-Stokes equations in primitive variables on Cartesian grids. Two numerical approaches were proposed in this work, which are based on the method of lines process in conjunction with transfer of all the variables from the boundaries to the nearest uniform grid knots. Initial value problems for the systems of ordinary differential equations for pressure and velocity components were computed using the one-step backward-differentiation predictor-corrector method or the Galerkin-Runge-Kutta method of third order. Some test calculations for laminar flows in square, half-square, triangular, semicircular, cubic, half-cubic, half-cylinder and hemisphere cavities with one uniform moving wall were reported. The present results were compared with the available data in the literature and the Fluent solver numerical simulations.

3

Finite difference schemes and the method of lines for solving incompressible viscous flow problems

Kosma Z., Motyl P.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 1

15-35

EN

For the determination of viscous incompressible flows a pure stream-function formulation for the fourth-order equation, the artificial compressibility method, and velocity correction method are employed. Test calculations are performed for various flows inside square, triangular, semicircular and cubic cavities with one uniform wall, the backward-facing step, double bent channels, the flow around an aerofoil at large angle of attack and for flows over models of buildings. Some complex geometrical configurations can be decomposed into a set of simpler subdomains. A practical methodology for the computation of the Navier-Stokes equations in arbitrarily complex geometries is also considered. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots.

4

The finite difference scheme for the solution of one quasilinear equation

Ciegis R., Ciegis R., Rate P.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

723-732

EN

In this paper we are interested in the solution of one-dimensional quasi-linear diffusion-reaction equation. The nonlinear reaction term includes the first derivative in space of the solution. We use the finite difference method to discretize this problem. The modification of a general methodology for investigation of difference schemes approximating non-stationary differential equations is used and the results for the stability and convergence of the numerical solution are proved. The convergence of the discrete derivative of the solution is proved in the maximum norm.