Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI

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