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Numerical Analysis

Numerical Scheme

The numerical scheme for which we have proven the analytical results below, and which we used for our numerical simulations is given by

\frac{\rho_i^{(n+1)} -\rho_i^{(n)}}{\Delta t} = \frac{1}{h^2}\left(D^2\rho_i^{(n+1)}-\rho_i^{(n+1)}D^2c_i^{(n+1)}-\left(Dc_{i-1}^{(n+1)}\right)^+D\rho_{i-1}^{(n+1)}+\left(Dc_{i}^{(n+1)}\right)^-D\rho_{i}^{(n+1)}\right)

and

\frac{D^2c^{(n+1)}_{i}}{h^2}+\rho_i^{(n+1)}-c_i^{(n+1)}=0

where

D^2u_i=u_{i+1}-2u_i+u_{i-1},\qquad Du_i=u_{i+1}-u_i

We proved that such a scheme is consistent with the Parabolic-Elliptic Keller-Segel equations. The statement of the theorem is as follows


Consistency 1

Convergence

We proved two major results for the numerical scheme given above which together give convergence of the scheme to a weak solution of the Keller-Segel equations. The first involves proving that the scheme has a unique solution at each time step. For this we proved the following theorem, which also gives us conservation of mass, positivity of the solution and a uniform bound on the discrete H^1 norm of the approximation to the chemical concentration c. The proof of existence involves a linearisation of the scheme together with an application of the Brouwer fixed point theorem and was adapted from the proof given by Filbet in the finite volume setting [1]. In the following we have made use of the trapezium rule, so for notational convenience we define the ''trapezium sum" as follows

\sum_{i=0}^{N}\mathop{}{\mkern-23mu \bullet}\: a_i = \frac{a_0+a_N}{2} + \sum_{i=1}^{N-1} a_i.

The statement of the first theorem is

Existence 1

The second result was split into two parts, the first of which was to prove convergence in some sense of the unique solution to the scheme to a couple (\rho,c) as the time step \Delta t and grid size h approach zero, and finally we proved that this (\rho,c) is a weak solution to the Keller-Segel equations. For this we first defined discrete functions \rho_h and c_h associated to the solution of the scheme as piecewise constant functions taking corresponding values at grid points and discrete time steps. We also define piecewise constant approximations d\rho_h and dc_h to the derivatives with d\rho_h(t,x) =\frac{\rho_{i+1}^{(n)}-\rho_i^{(n)}}{h} on (t^n,t^n]\times[x_i,x_{i+1}). We then needed some a-priori estimates on discrete norms of \rho_h and c_h and due to the form of the non-linearity in the parabolic equation, in order to pass to the limit we required strong convergence of \rho_h to \rho but only weak convergence of c_h to c and of the approximations to the derivatives, d\rho_h and dc_h to \rho_x and c_x. This result can be summarised in the following theorem, the proof of which was mostly adapted from [1].

Convergence 1


References

[1] F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, (2005)