High performance algorithms based on a new wavelet expansion for time dependent acoustic obstacle scattering

We remind that the solution of the scattering problem considered is given by the following series expansion in spherical harmonics:

u^s( x) = -4p

+¥
å
l = 0

i^l j_l(w^*)h_l⁽¹⁾(w^*|| x||)

h_l⁽¹⁾(w^*)

l
å
m = -l

Y_l^m(

^
x

)Y_l^m( a^*),

x = || x||

^
x

, || x || > 1,

^
x

Î ¶B,

(1)

where a^*=(0,0,-1)^T, w^*=1, h_l⁽¹⁾(z), j_l(z), z Î C, l = 0,1,¼ are the spherical Hankel and Bessel functions of index l respectively, Y_l^m, l = 0,1,¼, m = 0,±1,¼,±l, are the spherical harmonics and ¶B={x ÎR³ | || x ||=1 }.

We have computed the relative error committed when we replace the numerical solution u^s_w^*,S,m^*,t obtained with the numerical method proposed in the paper associated to this website with the (approximate) solution u^s_{L_max} obtained truncating the expansion (1) at l = L_max and choosing L_max = 10 on the following points on the boundary of a sphere having radius 1.5 and center the origin:

x_i,j^* = (1.5sin(

i)cos(

j),1.5sin(

i)sin(

j),1.5cos(

i))^T,

i = 1,2,¼,19, j = 0,¼,20 and i = 0,j = 1 and i = 20, j = 1,

(2)

We define the relative error as follows:

relative error =

(3)

é
ê
ê
ê
ê
ê
ë

å
i = 0,20

|u^s_w^*,S,m^*,t( x_i,1^*)-u^s_{L_max}( x_i,1^*)|²+

19
å
i = 1

20
å
j = 0

|u^s_w^*,S,m^*,t( x_i,j^*)-u^s_{L_max}( x_i,j^*)|²

å
i = 0,20

| u^s_{L_max}( x_i,1^*)|²+

19
å
i = 1

20
å
j = 0

|u^s_{L_max}( x_20,1^*)|²

ù
ú
ú
ú
ú
ú
û

^1/2

The approximate solution u^s_{L_max} when L_max = 10 on the sphere of center the origin and radius r = 1.5 is 4-6 digits accurate.

The results obtained are shown in the following table:

Table 1: Comparison between the solution computed with the method proposed and the spherical harmonics expansion

Number of (complex) unknowns used to approximate the solution of the integral equations	relative error
256	7.40e-02
1024	1.67e-02
4096	5.77e-03
16384	3.85e-04
65536	3.83e-04
262144	3.79e-04