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Convergent difference schemes for the Hunter–Saxton equation. (English) Zbl 1114.65101
The authors propose and analyze some finite difference schemes for the Hunter-Sexton equation \[ u_t + u u_x =\frac12 \int_0^x (u_x)^2 \,dx,\quad \;x > 0, \;t >0. \] This equation has been suggested as a simple model for nematic liquid crystals. They prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of \(u\), which is achieved by analyzing various renormalizations (in the sense DiPerna and Lions) of the numerical schemes. The authors demonstrate through some numerical example the proposed schemes as well as other schemes which have no rigorous convergence.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35A35 Theoretical approximation in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
76A15 Liquid crystals
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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