A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.