Diffusive Realization of the Impedance Operator on Circular Boundary for 2D Wave Equation
A new formulation of the perfectly matched operator on circular boundary for 2D wave equation is introduced. It is based on the “diffusive representation”, useful for a wide class of causal operators and which enables exact and easily approximable time-lo
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Summary. A new formulation of the perfectly matched operator on circular boundary for 2D wave equation is introduced. It is based on the "diffusive representation", useful for a wide class of causal operators and which enables exact and easily approximable time-local realisations of dissipative nature.
1 Introduction Consider the open unit disk in lR 2 , denoted by D, with boundary aD parameterized bye, as seen in Fig. 1. We propose to realize the impedance operator Z+ such that the solution u of the Cauchy problem with null initial conditions: a;u-L1u=f on(t,x,Y)ElR~xD { (1) u = z+avu on (t,x,y) E lR~ x aD be identical in D to the solution U of the same problem but in free space:
{ a;u -
supp U
L1U
c
on (t, x, y) E lR~ influence domain of f.
=f
X
lR 2
(2)
In such a case, not any reflexion is generated at the boundary: operator Z+ makes aD perfectly absorbing; it is a t-causal non local operator on S'(lRt x aD) (see for example [1]).
Spatial support of excitationj \.
y
t
. ... 1
~
x
aD Fig.!. The domain of problem (1)
G. C. Cohen et al. (eds.), Mathematical and Numerical Aspects of Wave Propagation WAVES 2003 © Springer-Verlag Berlin Heidelberg 2003
Diffusive Realization of the Impedance Operator
137
We show in the following sections that Z+ can be concretely realized by a specific equation (with null initial condition) of diffusive nature (in a sense specified later) with suitable dissipative properties. More precisely, by suitable choice of time-invariant spatial operators A, B, C (determined later) with etA strongly dissipative, we have:
Discretizing (3) following standard methods will lead to implement able numerical approximations of Z+ via finitedimensional 3 differential systems (e, w-discretization only), in such a way that:
x - AX = B OvUl8D
on JR~ x {el}z=l:L x {7Jdk=l:K
=}
Z+ OvUl8D c:::
ex.
(4)
Thanks to the specific properties of equation (3) (inherited from its diffusive nature), efficient approximations can be performed with small K (about 1020), which allows to realize numerical absorbing conditions of low cost.
2 Diffusive representation The concept of diffusive representation has been introduced for dynamical realizations of causal integral operators [2]. Absorbing control problems for propagative equations has already been successfully solved following this approach [3]. We recall the basic principle. Consider 1i a causal operator defined by: v f---+ Loo h(.-s) v(s) ds, h with moderated growth. We denote vt(s) := l]-oo,t](s) v(s), s E JR, the restriction of v to its past. With H = £h the Laplace Transform 4 of h and thanks to causality of 1i, (1iv) (t) does not depend, at any t, on the "future" of v (i.e.: (1i(v - vt)) (t) = 0); so we have for any v with left-bounded support:
(1iv) (t) = [£-1 (H £v)] (t) = [£-1 (H £vt)] (t).
(5)
Furthermore, H is analytic in JR~ + iJR and in several cases, the analyticity domain [2 is suc
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