Modified arc-length adaptive algorithms for degenerate reaction-diffusion equations.
The authors consider the following reaction-diffusion equation \[ u_t= \delta^2 u_{xx}+ p(x) u_x+ f(u),\quad 0< x< 1,\quad t> t_0 \] under homogeneous initial and Dirichlet boundary conditions, where \(0<\delta\leq 1\), \(0\leq p(x)\leq b\), \(0\leq x\leq 1\), \(t\geq t_0\). The singular source term \(f(u)\) is monotonically increasing for \(0\leq u< 1\) and \(f(0)= 1\), \(\lim_{u\to r}\, f(u)= \infty\). Modified monitor functions based on arc-lengths of the rate function \(u_t\) are constructed and used to obtain an adaptive semi-discretization on a non-uniform space grid and a two-stage fully adaptive temporal scheme. It is shown, under some assumptions on the step-size, that the numerical solution well imitates monotonic properties of the physical solution. A numerical example for \(p\equiv 0\), \(f(u)= (1- u)^{-\theta}\), \(\theta> 0\) is investigated in details.
scientific article; zbMATH DE number 1861023