Construction of space-time high resolution second order accurate three dimensional MP difference schemes

In this paper, a new Riemann-solver-free class of difference schemes are constructed to scalar nonlinear hyperbolic conservation laws in the three dimension (3D). We proved that these schemes had second order accuracy in space and time, and satisfied maximum principles (marked as MPs) under an appropriate CFL condition. This results in a second-order accuracy, MP schemes a natural extension of the one (two)-dimensional second-order. In addition, these schemes can still be extended to the vector system of conservation law. We yet prove that these schemes satisfied the scalar and vector maximum principle, and in the more general context of systems.