Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)

The paper deals with the Cauchy problem for the linear constant-coefficient strongly hyperbolic system $u_t+Au_x=({1}/{delta})B$.

The critical case where $B$ has a nontrivial nullspace is investigated. Under suitable assumptions on the matrices $A$ and $B$ the convergence in $L_2$ as $delta o 0$ of the solution $u(·,t,delta)$ of the Cauchy problem is proved. Then the evolution of the limit function ${check u}(·,t)$ as a solution of the Cauchy problem for the strongly hyperbolic system without zero-order term is described. This system is called an equilibrium system and a method for obtaining it is given. A concept of stiff well-posedness is introduced that ensures solution estimates independent of $0<deltale 1$. It is shown that for $2 imes 2$ systems the requirement of stiff well-posedness is equivalent to the well-known subcharacteristic condition. The theory is illustrated by examples. One of them shows that the assumption of strong hyperbolicity is essential. Lastly, some numerical experiments for a $3 imes 3$ system are carried out.