We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws, we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in earlier work, we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

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To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. In the language of dagger spaces introduced by Grosse-Kl\"onne, the $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of meromorphic differential forms with logarithmic poles for an affinoid toric dagger space over the dagger unit polydisc. It is a complex of ${\mathcal O}^\dagger$-modules with integrable connections and with Frobenius structures defined on the dagger unit polydisc such that traces of Frobenius on fibers at Techm\"uller points define the hypergeometric function over the finite field introduced by Gelfand and Graev.

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