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 />
Finite volume scheme for the problem...<br />
In addition, we take a usual two-point monotone consistent flux<br />
approximation to produce a discrete operator (divc F) T (·) which<br />
approximates the convection operator div F(·).<br />
With the notation introduced above, our discretization writes:<br />
find a discrete function u T,∆t satisfying for n = 1,...,N = T/∆t the <strong>equations</strong><br />
<br />
<br />
u<br />
<br />
<br />
<br />
T,n − u T,(n−1)<br />
+(div c<br />
∆t<br />
F) T [u T,n ]−div T [a0(∇ T w T,n )] = 0,<br />
w T,n = A(u T,n ),<br />
together with the boundary <strong>and</strong> initial conditions<br />
<br />
u<br />
for all n = 1,...,N,<br />
n K = 0 for all K near ∂Ω<br />
u n K ∗ = 0 for all K∗ near ∂Ω;<br />
u 0 K = 1<br />
<br />
<br />
mK<br />
K<br />
u0<br />
for all K, u 0 1<br />
K<br />
∗ =<br />
mK ∗<br />
K∗ u0 for all K ∗ .<br />
Theorem (main result of : Andr. & Bendahmane & Karlsen JHDE’11)<br />
The discrete solutions u T,∆t exist <strong>and</strong> converge to the unique entropy<br />
solution u as the discretization step (space <strong>and</strong> time) goes to zero.