Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems

Abstract

Consider a general strictly hyperbolic, quasilinear system, in one space dimesion $$u_t+A(u) u_x=0,\quad\quad\quad\quad\quad\quad(1)$$ where \({u \mapsto A(u), u\in\Omega\subset{\mathbb{R}}^N}\) , is a smooth matrix-valued map. Given an initial datum u(0, ·) with small total variation, let u(t, ·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as a limit of solutions to the viscous parabolic approximation ut + A(u)ux = μuxx, as μ → 0. For every T ≥ 0, we prove the a-priori bound $$\left\|u^{\varepsilon}(T,\cdot)-u(T,\cdot)\right\|_{{\mathbb{L}^1}}=o(1)\cdot\sqrt{\varepsilon}\,|\log{\varepsilon}|\quad\quad\quad\quad\quad\quad(1)$$ for an approximate solution \({u^{\varepsilon}}\) of (1) constructed by the Glimm scheme, with mesh size \({\Delta x = \Delta t = {\varepsilon}}\) , and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws ut + F(u)x = 0 satisfying the classical Lax or Liu assumptions on the eigenvalues λk(u) and on the eigenvectors rk(u) of the Jacobian matrix A(u) = DF(u).