Note that the desired state consists of shock states. The numerical results for this example are presented in Figure 2.

In the numerical results the relaxation method recovers the discontinuities and achieves optimality. A comparison of both approaches beginning with the case of smooth data in the domain [0, 1] x [0, T ] will be undertaken next. Again optimal initial data such that the flow properties at time T match the desired flow properties at time T given by the initial data. The optimization algorithm will be initialised with the initial profile:. The initial state computed by the optimal control approaches initial , target and optimized states for the LBE and the relaxation method are displayed in Figure 3.

Both methods converge to the target state and optimality is achieved. This example deals with the inverse design of a flow in a 1D shock-tube. Given a set of measurements on some actual flow at time T , the best estimate for the initial state that leads to the observed flow is determined. This problem has been explored before by many authors [30, 28, 29], but here, the linearization of the flow equation as described above and the underlying derivation of the adjoint variables for the computation of the gradient of the objective functional is used.

An example taken from [34] is considered. The desired state is the solution of the Riemann problem with the initial data. The state consists of different smooth and non-smooth waves. The initial guess for the iterative optimization problem is chosen to be. The results, using the adjoint method combined with the Lattice-Boltzmann and the relaxation approaches are presented in Figure 4.

The initial control u 0 , the desired state u d , the optimized state and solution of the optimized flow at time T are reported in Figure 5 and it can be observed that the discontinuity is well recovered.

## Figure 1 from Worksheet 5 : Shallow Water Equations in CUDA T 5 . 1 : Case Study - Semantic Scholar

It can also be observed that the discontinuous initial data yields several waves in the solution. The results reveal that the approaches presented here, using the Lattice-Boltz-mann model or the relaxation method, are both able to recover solutions with discontinuities such as shocks, rarefactions or contact discontinuities. In the following, the convergence behaviour of the schemes presented in this paper is analysed. Precisely, the convergence of the optimization algorithm using the Lattice Boltzmann and the relaxation approaches in terms of the number of grid points will be studied.

Hence the solution for the discontinuous example presented in Section 5.

## Finite volume methods for hyperbolic problems

In Table 1 the number of iterations obtained with the first and second order scheme for the lattice Boltzmann method, and with a first order scheme for the relaxation method, and the CPU times until convergence are presented. Note that the number of optimization iterations are independent of the grid size N x.

In Figure 6 the evolution of the cost functional as a function of the number of optimization iterations is presented. The relaxation approach converges faster than the lattice Boltzmann approach. In this paper two approaches for the control of flows governed by a system of conservation laws have been presented namely: the Lattice-Botzmann approach and the relaxation method, both combined with an adjoint calculus algorithm for the optimization. The resulting numerical schemes perform convincingly, since they are able to handle flows with discontinuities such as shocks or contact discontinuities.

Moreover, for the Euler system, the extension of the optimization methods to higher dimensions can be done. Possible difficulties would come from the boundary conditions. The multidimensional model can be used effectively to solve further interesting problems as they arise in shape optimization or topology design.

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Castro, F. Palacios and E. Zuazua, An alternating descent method for the optimal control of the inviscid Burgers equation in the presence ofshocks. Models and Meth. Castro and E. Zuazua, Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids. AIAA Journal, 45 , Baeza, C. Carlos, F. Zuazua, 2D Euler shape design on non-regular flows using adjoint Rankine-Hugoniot relations. AIAA Journal, 47 , Hager, Runge-Kutta methods in optimal control and the transformed adjoint system.

Numerische Mathematik, 87 , Chikatamarla and I. Karlin, Lattices for the lattice Boltzmann method. Physical Review, E79 Oxford University Press Chen and G. Doolen, Lattice Boltzmann method for fluid flows. Fluid Mech. Banda and M. Seaid, Higher-order relaxation schemes for hyperbolic systems of conservation laws. Math, 13 3 , Methods Partial Differ. Equations, 23 5 , Jin and Z. If you have any problems with your order please contact us first before leaving feedback and our excellent customer service help to resolve the issue.

We offer a 30 day no quibble money back guarantee. Payments We currently only accept immediate payment by PayPal for all eBay orders. Payment must be received before items can be despatched. Western Europe deliveries are expected to arrive between business days. Finite Volume Methods for Hyperbolic Problems. Randall J. LeVeque , Leveque Randall J. Cambridge University Press , This book contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws.

These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline.