Partial Differential Equations - Modelling and ... - ResearchGate

68 J.Ch. Gilbert and P. Joly
tends to 0 – typically the step size of the computational mesh) of problems of
the form:
d 2 u h
dt 2 + A hu h =0, (2)
where the unknown u h is a function of time with value in some Hilbert space V h
(whose norm will be denoted ‖·‖, even if it does depend on h) andA h denotes
a bounded self-adjoint and positive operator in V h (namely an approximation
of the second order differential operator A). Several approaches lead naturally
to problems of the form (2), among which
• variational finite differences [CJ96, Dab86, AKM74],
• finite element methods [CJRT01, CJKMVV99],
• mixed finite element methods [CF05, PFC05],
• conservative discontinuous Galerkin methods [HW02, FLLP05].
Of course, the norm of A h blows up when h goes to 0, as
‖A h ‖ = O(h −2 ).
It is well known that one has conservation of the discrete energy:
E h (t) = 1 2
du h
2 ∥ dt ∥ + 1 2 a h(u h ,u h ),
where a h (·, ·) is the continuous symmetric bilinear form associated with A h .
From the energy conservation result and the positivity of A h , one deduces a
stability result: the norm of the solution u h (t) can be estimated in function
of the norm of the Cauchy data:
u 0,h = u h (0), u 1,h = du h
dt (0),
with constants independent of h. This is also a direct consequence of the
formula:
[ ] [
]
u h (t) = cos A 1 2
h
t u 0,h + A − 1 2
h
sin A 1 2
h
t u 1,h ,
which yields
‖u h (t)‖ ≤‖u 0,h ‖ + t‖u 1,h ‖. (3)
In what follows, we are interested in the time discretization of (2) by explicit
finite difference schemes. More specifically, we are interested in the stability
analysis of such schemes, i.e., in obtaining a priori estimates of the form
(3) after time discretization. The conservative nature (i.e., the conservation
of energy) of the continuous problem can be seen as a consequence of the
time reversibility of this equation. That is why we shall favor centered finite
difference schemes which preserve such a property at the discrete level.
The most well known scheme is the classical second order leap-frog scheme.
Let us consider a time step ∆t > 0 and denote by u n h ∈ V h an approximation
of u h (t n ), t n = n∆t. Thisschemeis