Parameter identification in a semilinear hyperbolic system

Abstract

We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.

Efficient numerical methods for gas network modeling and simulation

“…For the well-posedness and the regularity of the solution of the system (5)-(6), we refer to [9]. More advanced numerical schemes for hyperbolic PDEs, such as the total variation diminishing (TVD) method [40], or the discontinuous Galerkin method (DG) [24], could be implemented at the next step of our research to investigate more complicated dynamics of the gas networks.…”

“…Note that the nonlinear term q|q| p which describes the damping law of gas networks is still differentiable with respect to the change of the flow direction. For details of the regularity of the solution (p, q) of the PDE (5), we refer to Theorem 3.2 given in [9]. It is also pointed out by Theorem 4.3 in [9] that the forward PDE operator of (5) is Fréchet differentiable with Lipschitz continuous derivative.…”

“…For details of the regularity of the solution (p, q) of the PDE (5), we refer to Theorem 3.2 given in [9]. It is also pointed out by Theorem 4.3 in [9] that the forward PDE operator of (5) is Fréchet differentiable with Lipschitz continuous derivative. Therefore, we can still apply Newton's method to solve such a nonlinear system of equations.…”

Series solutions of a semilinear wave equation

“…We are concerned with the construction of special series solutions of a classical one dimensional semilinear wave equation defined by Equation models the damping vibration of a string and also the propagation of pressure waves in a gas pipeline, where λ u 2 accounts for the friction against the pipe walls. The nonlinearity is known to cause finite time blowup, for example is such a solution in case a =0 and λ = 6.…”

“…(1) Equation 1 models the damping vibration of a string and also the propagation of pressure waves in a gas pipeline, 1 where u 2 accounts for the friction against the pipe walls. The nonlinearity is known to cause finite time blowup, for example u(x, t) = 1 ( t− T) 2 is such a solution in case a = 0 and = 6.…”