Construction of solutions and \(L^{1}\)-error estimates of viscous methods for scalar conservation laws with boundary
The authors study an initial boundary value problem for a scalar conservation law \(u_t+f(u)_x=0\), \(t,x>0\) with strictly convex flux function \(f(u)\). This problem is assumed to admit piecewise smooth entropy solutions. The authors describe the structure of such solutions, in particular they show that new types of waves appear. Based on the structure of entropy solutions the authors derive new \(L^1\)~error estimates for the vanishing viscosity method. It turns out that the error of the viscosity solution to the inviscid one is bounded by \(O(\varepsilon^{1/2})\). In the case when the inviscid solutions has special structure (it does not contain waves of new types) the error estimate is \(O(\varepsilon| \ln\varepsilon| )\) as for the initial value problem.
scientific article; zbMATH DE number 5162317
scientific article