Download Acoustic wave equation

Acoustic wave equation
Helmholtz
Helmholtz (wave)
(wave) equation
equation (time-dependent)
(time-dependent)
-- Regular
Regular grid
grid
-- Irregular
Irregular grid
grid
Numerical
Numerical Examples
Examples
Numerical Methods in Geophysics
Finite Elements
The Acoustic Wave Equation 1-D
How do we solve a time-dependent problem such
as the acoustic wave equation?
∂ t2u − v 2 ∆u = f
where v is the wave speed.
using the same ideas as before we multiply this equation with
an arbitrary function and integrate over the whole domain, e.g. [0,1], and
after partial integration
1
1
1
0
0
0
2
2
∂
u
ϕ
dx
−
v
∫ t j
∫ ∇u∇ϕ j dx = ∫ fϕ j dx
.. we now introduce an approximation for u using our previous
basis functions...
Numerical Methods in Geophysics
Finite Elements
The Acoustic Wave Equation 1-D
N
u ≈ u~ = ∑ ci (t )ϕ i ( x)
i =1
note that now our coefficients are time-dependent!
... and ...
N
∂ t2u ≈ ∂ t2u~ = ∂ t2 ∑ ci (t )ϕ i ( x)
i =1
together we obtain
1
1
1
⎤
⎡
⎤
⎡
2
2
c
ϕ
ϕ
dx
v
+
∂
⎢∑ ci ∫ ∇ϕ i ∇ϕ j dx ⎥ = ∫ fϕ j
⎢∑ t i ∫ i j ⎥
0
⎦ 0
⎣ i 0
⎦
⎣ i
which we can write as ...
Numerical Methods in Geophysics
Finite Elements
Time extrapolation
1
1
1
⎡
⎤
⎡
⎤
2
2
∂
+
c
ϕ
ϕ
dx
v
⎢∑ t i ∫ i j ⎥
⎢∑ ci ∫ ∇ϕ i ∇ϕ j dx ⎥ = ∫ fϕ j
0
⎣ i
⎦
⎣ i 0
⎦ 0
M
mass matrix
A
stiffness matrix
b
... in Matrix form ...
M T c + v 2 AT c = g
... remember the coefficients c correspond to the
actual values of u at the grid points for the right choice
of basis functions ...
How can we solve this time-dependent problem?
Numerical Methods in Geophysics
Finite Elements
FD extrapolation
M T c + v 2 AT c = g
... let us use a finite-difference approximation for
the time derivative ...
⎛ ck +1 − 2c + ck −1 ⎞ 2 T
M ⎜
⎟ + v A ck = g
2
dt
⎝
⎠
T
... leading to the solution at time tk+1:
[
]
ck +1 = ( M T ) −1 ( g − v 2 AT ck ) dt 2 + 2ck − ck −1
we already know how to calculate the matrix A but
how can we calculate matrix M?
Numerical Methods in Geophysics
Finite Elements
The mass matrix
1
1
1
⎡
⎤
⎡
⎤
2
2
c
ϕ
ϕ
dx
v
+
∂
⎢∑ ci ∫ ∇ϕ i ∇ϕ j dx ⎥ = ∫ fϕ j
⎢∑ t i ∫ i j ⎥
0
⎣ i 0
⎦ 0
⎣ i
⎦
... let’s recall the definition of our basis functions ...
x
⎧ ~
⎪h +1
⎪ i −1 ~
x
⎪
ϕ i ( ~x ) = ⎨ 1 −
hi
⎪
⎪ 0
⎪⎩
1
M ij = ∫ ϕ iϕ j dx
0
i=1
+
2
+
h1 h2
3
+
4
+
5
+
h3 h4
Numerical Methods in Geophysics
6
+
h5
7
+
for
− hi −1 < ~
x ≤0
for
0<~
x < hi
,~
x = x − xi
elsewhere
... let us calculate some
element of M ...
h6
Finite Elements
The mass matrix – some elements
Diagonal elements: Mii, i=2,n-1
1
hi −1
0
0
M ii = ∫ ϕ iϕ i dx =
∫
2
hi −1 hi
=
+
3
3
i=1
+
2
+
h1 h2
3
+
4
+
2
⎛ x ⎞
⎛
x⎞
⎜⎜
⎟⎟ dx + ∫ ⎜⎜1 − ⎟⎟ dx
hi ⎠
⎝ hi −1 ⎠
0⎝
hi
for
x ≤0
− hi −1 < ~
for
0<~
x < hi
elsewhere
ϕi
5
+
h3 h4
6
+
h5
7
+
xi
h6
hi-1
Numerical Methods in Geophysics
x
⎧ ~
⎪h +1
⎪ i −1 ~
x
⎪
ϕ i ( ~x ) = ⎨ 1 −
hi
⎪
⎪ 0
⎪⎩
hi
Finite Elements
Matrix assembly
Mij
Aij
% assemble matrix Mij
% assemble matrix Aij
M=zeros(nx);
A=zeros(nx);
for i=2:nx-1,
for i=2:nx-1,
for j=2:nx-1,
for j=2:nx-1,
if i==j,
if i==j,
M(i,j)=h(i-1)/3+h(i)/3;
A(i,j)=1/h(i-1)+1/h(i);
elseif j==i+1
elseif i==j+1
M(i,j)=h(i)/6;
A(i,j)=-1/h(i-1);
elseif j==i-1
elseif i+1==j
M(i,j)=h(i)/6;
A(i,j)=-1/h(i);
else
else
M(i,j)=0;
A(i,j)=0;
end
end
end
end
Numerical Methods in Geophysics
end
end
Finite Elements
Numerical example – regular grid
Numerical Methods in Geophysics
Finite Elements
Implicit Time extrapolation
Let is recall the ODE:
dT
= f (T , t )
dt
Before we used a forward difference scheme, what happens
if we use a backward difference scheme?
T j − T j −1
dt
+ O(dt ) = f (T j , t j )
⇒ T j ≈ T j-1 + dtf (T j , t j )
Numerical Methods in Geophysics
Finite Elements
Implicit schemes - stability
or
T j ≈ T j −1 (1 +
T j ≈ T0 (1 +
dt
τ
dt
τ
) −1
)− j
Is this scheme convergent?
Does it tend to the exact solution as dt->0? YES, it does (exercise)
Is this scheme stable, i.e. does T decay monotonically? This requires
1
0<
<1
dt
1+
τ
Numerical Methods in Geophysics
Finite Elements
What is an implicit method?
Let is recall the ODE:
dT
= f (T , t )
dt
Before we used a forward difference scheme, what happens
if we use a backward difference scheme?
T j − T j −1
dt
+ O(dt ) = f (T j , t j )
⇒ T j ≈ T j-1 + dtf (T j , t j )
Numerical
Methods
in Geophysics
Numerical Methods
in Geophysics
Implicit
FiniteMethods
Elements
FD extrapolation - implicit
M T c + v 2 AT c = g
... let us use an implicit finite-difference approximation for
the time derivative ...
⎛ ck +1 − 2c + ck −1 ⎞ 2 T
M ⎜
⎟ + v A ck +1 = g
2
dt
⎝
⎠
T
... leading to the solution at time tk+1:
[
2
2
] (gdt
T −1
ck +1 = M + v dt A
T
2
)
+ M T (2c − ck −1 )
How do the numerical solutions compare?
Numerical Methods in Geophysics
Finite Elements