To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws to convection diffusion equations, we propose a non-conventional high order finite volume weighted essentially non-oscillatory (WENO) scheme which can be proven maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.

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Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.

Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under dissolution of injected acid. The important physical features of porosity and acid concentration include their boundedness between $0$ and $1$, as well as the monotone increasing for porosity. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In this paper, we propose high-order bound-preserving discontinuous Galerkin methods to keep these important physical properties. The main technique is to introduce a new variable $r$ to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time for $r$. Moreover, the high-order accuracy is attained by the flux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique.