Explicit formula for the limit of a difference approximation
The conservation law \(u_ t+(\log (a \exp (a)+b \exp (-a)))_ x=0\) with \(a>0\), \(b>0\), \(a+b=1\), \(u(x,0)=u_ 0(x)\) is considered in the quarter plane \(x>0\), \(t>0\). An approximate solution of this problem is constructed using the difference scheme \(u_ k^{n+1}=u^ n_ k+(\Delta t/\Delta x)(g(u^ n_{k-1},u^ n_ k)-g(u^ n_ k,u^ n_{k+1})),\) \(u^ 0_ k=u_ 0(k\Delta x),\quad u^ n_ 0=\lambda (n\Delta t),\) where \(g(u,v)=\log (a \exp (u)+b \exp (v)).\) The main result of the paper consists in obtaining an expression for the solution of the difference equations and its limit when the temporal and spatial steps tend to zero.
scientific article; zbMATH DE number 4178681