Implicit-Explicit Runge-Kutta schemes for hyperbolic systems ... - utenti

Space discretizations
We consider the case of the single scalar equation
ut + f(u)x = 1
ε g(u).
We have to distinguish between schemes based on cell averages (finite volume approach
as in most schemes) and schemes based on point values (finite difference approach).
Let ∆x and ∆t be the mesh widths. We introduce the grid points
xj = j∆x, xj+1/2 = xj + 1
∆x,
2
j = . . . , −2, −1, 0, 1, 2, . . .
and use the standard notations
Finite volumes
u n j = u(xj, t n ), ū n j = 1
∆x
� xj+1/2
xj−1/2
u(x, t n ) dx.
Integrating the equation on Ij = [x j−1/2, x j+1/2] and dividing by h we obtain
�
dū�
�
dt
� j
= − 1
∆x [f(u(x j+1/2, t)) − f(u(x j−1/2, t)) + 1
ε∆x g(u)|j
As usual the key step is the reconstruction step necessary to reconstruct the function
u(x, t) at the grid points (required to evaluate the right hand side) starting from its cell
average u(x, ¯ t).
26