Rarefactions and large time behavior for parabolic equations and monotone schemes
Solutions of the degenerate parabolic equation \(u_ t+f(u)_ x=A(u)_{xx}\) and of the associated difference equation \[ u^{n+1}(x)- u^ n(x)=-\lambda \Delta_ d(g(u^ n(x-p_ 0d),...,u^ n(x+q_ 0d)) \] are approximated for large t by those of \(u_ t+f(u)_ x=0\). Here u is scalar, f is convex and A'(u)\(\geq 0\). For \(A(u)=| u|^{\gamma}u\) we have the porous medium equation and for \(f(u)=0,5u^ 2\), \(A(u)=u\) we get Burger's equation. An asymptotic convergence rate is also given.
scientific article; zbMATH DE number 4052921