In this article, we consider a new technique that allows us to overcome the well-known restriction of Godunov's theorem. According to Godunov's theorem, a second-order explicit monotone scheme does not exist. The techniques in the construction of high-resolution schemes with monotone properties near the discontinuities of the solution lie in choosing of one of two high-resolution numerical solutions computed on different stencils. The criterion for choosing the final solution is proposed. Results of numerical tests that compare with the exact solution and with the numerical solution obtained by the first-order monotone scheme are presented.

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