Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik An operator splitting method for nonlinear convection-diffusion equations. (English) Zbl 0882.35074 Numer. Math. 77, No. 3, 365-382 (1997). Summary: We present a semi-discrete method for constructing approximate solutions to the initial value problem for the \(m\)-dimensional convection-diffusion equation \(u_{t}+\nabla\cdot \vec F(u) =\varepsilon\Delta u\). The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case \(m>1\), dimensional splitting is used to reduce the \(m\)-dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. Cited in 1 ReviewCited in 26 Documents MSC: 35L65 Hyperbolic conservation laws 35A35 Theoretical approximation in context of PDEs 35K15 Initial value problems for second-order parabolic equations Keywords:semi-discrete method; dimensional splitting; fully discrete method PDFBibTeX XMLCite \textit{K. H. Karlsen} and \textit{N. H. Risebro}, Numer. Math. 77, No. 3, 365--382 (1997; Zbl 0882.35074) Full Text: DOI