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We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

In this paper, we discuss high order finite difference weighted essentially non-oscillatory (WENO) schemes, coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintanance of third order spatial / temporal accuracy when the limiters are applied to a third order finite difference scheme and third order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth order accuracy.

We present novel algorithms for compressible flows that\n\nare efficient for all Mach numbers. The approach is based on several\n\ningredients: semi-implicit schemes, the gauge decomposition\n\nof the velocity field and a second order formulation of the density\n\nequation (in the isentropic case) and of the energy equation (in\n\nthe full Navier-Stokes case). Additionally, we show that our approach\n\ncorresponds to a micro-macro decomposition of the model,\n\nwhere the macro field corresponds to the incompressible component\n\nsatisfying a perturbed low Mach number limit equation and\n\nthe micro field is the potential component of the velocity. Finally,\n\nwe also use the conservative variables in order to obtain a proper\n\nconservative formulation of the equations when the Mach number\n\nis order unity. We successively consider the isentropic case, the\n\nfull Navier-Stokes case, and the isentropic Navier-Stokes-Poisson\n\ncase. In this work, we only concentrate on the question of the\n\ntime discretization and show that the proposed method leads to\n\nAsymptotic Preserving schemes for compressible flows in the low\n\nMach number limit.

In this paper, we present a mixed generalized multiscale finite element method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries, and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of