When solving convection dominated partial differential equations,such as the incompressible and compressible Euler equations in fluid dynamics, it is a challenge to design numerical schemes which are both strongly stable and high order accurate, especially when the solution contains sharp gradient regions or discontinuities. Previous schemes satisfying strict maximum principle for scalar equations and positivity-preserving for systems are mostly first order, or at most second order accurate. We construct uniformly high order accurate discontinuous Galerkin
(DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) schemes satisfying a strict maximum principle for scalar conservation laws and passive convection in incompressible flows, and positivity preserving for density and pressure for compressible Euler equations. One remarkable property of our approach is that it is
straightforward to extend the method to two and higher dimensions on arbitrary triangulations. We will also emphasize recent developments including arbitrary equations of state, source terms, integral terms, shallow water equations, high order accurate finite difference
positivity preserving schemes for Euler equations, and a special non-standard positivity preserving high order finite volume scheme for convection-diffusion equations. Numerical tests demonstrating the good performance of the scheme will be reported. This is a joint work with Xiangxiong Zhang.
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