Stepsize restrictions for total-variation-boundedness in general Runge--Kutta procedures
The Cauchy problem for the equation \[ {\partial u(x,t)\over\partial t}+ {\partial f(u(x,t))\over\partial x}= 0,\quad t\geq 0,\;-\infty< x<\infty, \] using the method of lines, can be regarded as an ordinary differential equation in the form \[ {dU(t)\over dt}= F(U(t)),\quad U(t_0)= u_0\tag{1} \] with \(U\in\mathbb{R}^\infty\). The authors apply the Runge-Kutta method to (1) and indicate a stepsize restriction \(\Delta t\leq\Delta t_0\), which guarantee total variation bounded or total variation diminishing for the approximations \(U_n\), obtained through this method. They use for the proof of the main theorem two coefficients \(R(A,b)\) and \(S(A,b)\) defined for an irreducible Runge-Kutta method. The practical calculation of these coefficients is difficult.
scientific article; zbMATH DE number 2160263
scientific article