A nonoscillatory numerical scheme based on a general solution to nonlinear advection-diffusion equations (Q2706029)

The paper is devoted to a nonoscillatory numerical method for nonlinear advection-diffusion equations proposed by \textit{K. Sakai} [J. Comput. Appl. Math. 108, No. 1-2, 145-156 (1999; Zbl 0935.65107)] for linear advection-diffusion equations. The authors consider two problems: The one-dimensional advection-diffusion equation NEWLINE\[NEWLINE\partial u/\partial t + u\partial u/\partial x =\nu \partial ^2u/\partial x^2NEWLINE\]NEWLINE (\(t\)-time, \(x\)-space, \(\nu \)-the diffusion coefficient, \(u\)-velocity in some computational region) and the two-dimensional advection-diffusion system NEWLINE\[NEWLINE\partial u/\partial t + u\partial u/\partial x + v\partial u/\partial y =\nu (\partial ^2u/\partial x^2+\partial ^2u/\partial y^2),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\partial v/\partial t + u\partial v/\partial x + v\partial v/\partial y =\nu (\partial ^2v/\partial x^2+\partial ^2v/\partial y^2)NEWLINE\]NEWLINE (\(u\) and \(v\) are components of the velocity in some computational region). The method is based on the spectral technique. Applying the Cole-Hopf transformation, the authors obtain a general solution to the initial value problem for the above stated equations. Based on this general solution, a numerical scheme for nonlinear unsteady advection-diffusion equations is established. A numerical example of propagation of a shock wave for the one-dimensional nonlinear Burgers equation is shown.