Solitary waves of the Korteweg–de Vries–Burgers' equation

Abstract A finite element solution of the Korteweg–de Vries–Burgers' equation (KdVB) based on Bubnov–Galerkin's method using cubic B-splines as element shape and weight functions, is set up. A linear stability analysis shows the scheme to be unconditionally stable. Simulations undertaken proved that the scheme can model faithfully the Korteweg–de Vries equation ( ν =0), Burgers' equation ( μ =0) as well as the Korteweg–de Vries–Burgers' equation ( ν , μ ≠0). Simulations studied included the solution of Burgers' equation for arbitrary initial condition, the migration of a single solitary wave, the temporal evaluation of a Maxwellian and the time evaluation of the solutions of the KdVB equation with various values for the diffusion and dispersion coefficients. Invariants and error norms are studies whenever possible to determine the conservation properties of the algorithm.