Solitary waves of the Korteweg-de Vries-Burgers' equation
The author studies the numerical solution of the singularly perturbed Korteweg-de Vries-Burgers equation in one spatial dimension, \[ u_t-\nu u_{xx}+\mu u_{xxx}+\varepsilon uu_x= 0,\quad x\in [a,b],\quad t>0,\quad \nu,\mu,\varepsilon> 0. \] For the spatial discretization, a finite element method using cubic B-splines as ansatz and test functions is proposed. Appearing integrals are approximated by means of a Gaussian quadrature formula. The discretization in time is a Crank-Nicolson scheme. The nonlinearity is resolved by an inner iteration. The method is shown to be unconditionally stable in the sense of von Neumann. Numerical tests are presented for the cases \(\nu= 0\) (Korteweg-de Vries equation), \(\mu=0\) and small \(\nu\) (Burgers equation), and \(\nu,\mu\neq 0\) (Korteweg-de Vries-Burgers equation). The test for the Burgers equation shows high accuracy. The numerical solution of the Korteweg-de Vries test equation also fulfills conservation principles known for the exact solution.
scientific article; zbMATH DE number 1462870