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 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 show 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.
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Received by the editor April 13, 2016, and, in revised form, April 28, 2016, October 17, 2016, and December 12, 2016. 2010 Mathematics Subject Classification. Primary 65M06, 65M12, 35Q92. The first author was partially supported by RNMS11-07444 (KI-Net) and NSF grant DMS 1514826. The second author was partially supported by a start-up fund from the State University of New York at Buffalo and NSF grant DMS 1620135. The third author was partially supported by a start-up fund from Peking University and RNMS11-07444 (KI-Net).