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| Partial differential equations arising in science and technology typically contain some structural information
reflecting important physical properties such as positivity, mass and energy conservation, or entropy dissipation.
Entropy dissipation is intensively used in the mathematical analysis of PDEs
for the derivation of apriori estimates which represent a cruical tool
in proving the existence of solutions and studing their long-time behaviour.
Our aim is, using the entropy structure of the cross-diffusion system from population dynamics,
to derive new numerical schemes which are structure-preserving.
The used techniques are based on translation of the continuous entropy estimates to the discrete level
with hope to obtain accurate and stable approximations.
We introduce new one-leg multistep time discretizations of considered cross-diffusion model for which
the existence of semi-discrete weak solutions is proved.
The main features of the scheme
are the preservation of the nonnegativity and the entropy-dissipation
structure. The key ideas are to combine Dahlquist's
G-stability theory with entropy-dissipation methods and to introduce a
nonlinear transformation of variables which provides a needed quadratic structure
in the equations.
It is shown that G-stability of the one-leg scheme is sufficient to derive
discrete entropy dissipation estimates. Furthermore,
under some assumptions on the evolution operator, the
second-order convergence of solutions is proved.
In order to underline the theoretical results, some numerical experiments
will be presented. |
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