Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
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<strong>Degenerate</strong> Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators <strong>and</strong> scheme Discrete calculus & Convergence analysis<br />
Discrete calculus tools...<br />
“Entropy” arguments: take (η ± c ) ′ (u T ) for test function, get<br />
Discrete entropy inequalities<br />
L ∞ bound (from comparison with constant solutions)<br />
We already know that the discrete convection operator with monotone<br />
flux leads to discrete entropy inequalities with remainder terms<br />
controlled by the “weak BV” estimate , Eymard, Gallouët, Herbin .<br />
In addition, we have to establish that div T a0(∇ T ·) “coexists nicely”<br />
with the entropy technique , i.e.,<br />
<br />
− div T k(∇ T A(u T ))∇ T A(u T ), θ(u T )ψ T<br />
<br />
(with θ = (η ± c ) ′ , ψT 0) behaves more or less like<br />
<br />
<br />
− div k(∇A(u))∇A(u)·(θ(u)ψ) := k(∇A(u))∇A(u)· ∇<br />
Ω<br />
Ω<br />
θ(u)ψ <br />
<br />
k(∇A(u)) θ(u)∇A(u) <br />
· ∇ψ = k(∇A(u))∇ Aθ(A(u))· ∇ψ<br />
Ω<br />
Here we need to replace the chain rule by a convexity inequality <strong>and</strong><br />
assume the orthogonality of the meshes.<br />
Ω