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.