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An operator splitting method for nonlinear convection-diffusion equations

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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.

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ReceivedFebruary 1, 1996 / Revised version received June 24, 1996

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Karlsen, K., Risebro, N. An operator splitting method for nonlinear convection-diffusion equations . Numer. Math. 77, 365–382 (1997).

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