The authors consider the following (possibly degenerate) equation \[ \partial_t u= \text{div}(f(u)\nabla(V(x)+ h(u))) \] admitting an entropy functional for \(x\in\Omega\), \(t> 0\), where \(\Omega\subset\mathbb{R}^d\) is an open bounded domain or \(\Omega=\mathbb{R}^d\), \(u\geq 0\), under the initial condition \(u(0,x)= u_0(x)\). (In the paper for simplicity the case \(\Omega= (a,b)\subset\mathbb{R}\) is considered.)
A finite volume scheme second-order accurate in space is constructed. – Numerical simulations are applied to the porous media equation, the drift-diffusion system for semiconductors, the Fokker-Planck equation for bosons and fermions and the Buckley-Leverett equation showing a preservation of steady states and long time behavior, also in the degenerate case.