Partial Differential Equations - Modelling and ... - ResearchGate

98 I. Sazonov et al.
In the first stage, the solution is advanced from time level t n to time level
t n+1/2 = t n + ∆t/2 using the forward difference approximation
U (n+1/2) = U (n) − ∆t
2
(
A k ∂U
∂x k
) (n)
. (6)
Here, the superscript (n) denotes an evaluation at time t = t n . In the second
stage, the solution at time level t n+1 = t n + ∆t is obtained from the central
difference approximation
(
U (n+1) = U (n) − ∆t A k ∂U ) (n+1/2)
. (7)
∂x k
At time t = t n , a continuous piecewise linear approximation, on element e,
may be expressed as
U (n)
e
= N (J) U (n)
(J) , (8)
where N (J) is the piecewise linear shape function associated with node J of the
mesh, U (J) represent nodal values and the implied summations extend over
each node J of element e. A variational formulation [ZM06] of the equation (6)
is employed to obtain the solution at time level t = t n+1/2 . To obtain the
solution at the end of the time step, at each node I, the weak variational
formulation [ZM06]
M (IJ) U (n+1)
(J)
= M (IJ) U (n)
(J) + Ak ∫
e∈Ω
∫
U e
(n+1/2) ∂N (I)
dΩ −
∂x k
Γ
˜F (n)
n N (I) dΓ
for the equation (7) is employed over the computational domain, Ω. Inthe
equation (9), Γ denotes the boundary of region Ω, ˜F n is a normal boundary
flux and M (IJ) is the standard consistent mass matrix for the mesh of linear
triangular elements in Ω. The equation (9) is solved by explicit iteration and
the resulting algorithm is stable provided that a CFL condition of the form
∆t ≤Cmin h e (10)
e
is satisfied, where h e denotes the minimum height of element e and C is a
safety factor.
For scattering simulations, the boundary condition at the surface of the
PEC scatterer is weakly imposed through the Galerkin statement. The truncated
far field boundary is taken to be rectangular in shape and a structured
grid of triangular elements is used to discretise the PML region.
(9)
4 A Co-Volume Method
For the co-volume method, the governing equations are considered in the
integral, time domain form of the equation (1) and the discretisation is accomplished
using two mutually orthogonal meshes [Mad95, GL93]. For this