Abstract
The biocompact difference scheme earlier proposed by these authors for a linear transport equation, which has the fourth-order approximation in spatial coordinate on the two-point stencil and the first-order approximation in time, is monotonic. This implicit scheme is absolutely stable and can be solved by explicit formulas of a running calculation. On the basis of this scheme a monotone non-linear homogeneous difference scheme of high (third for smooth solutions) order accuracy in time is constructed. Calculations of test problems with discontinuous solutions have demonstrated that the proposed scheme has a significant advantage in accuracy over the known nonoscillatory schemes of high-order approximation.
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