def step4(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
v = zeros_like(self.u)
for i in range(self.Nb+5, self.N-self.Nb-5):
L = self.calc_poly(i, 4)
r = i - L
if r == -1:
v[i] = 25.0/12.0*u_old[i-r+0] - 23.0/12.0*u_old[i-r+1] + 13.0/12.0*u_old[i-r+2] - 1.0/4.0*u_old[i-r+3]
elif r == 0:
v[i] = 1.0/4.0*u_old[i-r+0] + 13.0/12.0*u_old[i-r+1] - 5.0/12.0*u_old[i-r+2] + 1.0/12.0*u_old[i-r+3]
elif r == 1:
v[i] = -1.0/12.0*u_old[i-r+0] + 7.0/12.0*u_old[i-r+1] + 7.0/12.0*u_old[i-r+2] - 1.0/12.0*u_old[i-r+3]
elif r == 2:
v[i] = 1.0/12.0*u_old[i-r+0] - 5.0/12.0*u_old[i-r+1] + 13.0/12.0*u_old[i-r+2] + 1.0/4.0*u_old[i-r+3]
elif r == 3:
v[i] = -1.0/4.0*u_old[i-r+0] + 13.0/12.0*u_old[i-r+1] - 23.0/12.0*u_old[i-r+2] + 25.0/12.0*u_old[i-r+3]
else:
print "Error"
for i in range(self.Nb+6, self.N-self.Nb-6):
self.alpha = 1.01
#f[i] = 0.5*(self.f(v[i]) + self.f(v[i+1]) - self.alpha*(v[i+1]-v[i]))
f[i] = self.f(v[i])
for i in range(self.Nb+7, self.N-self.Nb-7):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def newstep(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
v = zeros_like(self.u)
for i in range(self.Nb+4, self.N-self.Nb-4):
L = self.calc_poly(i, 3)
r = i - L
if r == -1:
v[i] = 11.0/6.0*u_old[i-r+0] - 7.0/6.0*u_old[i-r+1] + 1.0/3.0*u_old[i-r+2]
elif r == 0:
v[i] = 1.0/3.0*u_old[i-r+0] + 5.0/6.0*u_old[i-r+1] - 1.0/6.0*u_old[i-r+2]
elif r == 1:
v[i] = -1.0/6.0*u_old[i-r+0] + 5.0/6.0*u_old[i-r+1] + 1.0/3.0*u_old[i-r+2]
elif r == 2:
v[i] = 1.0/3.0*u_old[i-r+0] - 7.0/6.0*u_old[i-r+1] + 11.0/6.0*u_old[i-r+2]
else:
print "Error"
for i in range(self.Nb+5, self.N-self.Nb-5):
self.alpha = 1.01
#f[i] = 0.5*(self.f(v[i]) + self.f(v[i+1]) - self.alpha*(v[i+1]-v[i]))
f[i] = self.f(v[i])
for i in range(self.Nb+6, self.N-self.Nb-6):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def step(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
for i in range(self.Nb, self.N-self.Nb):
L = self.calc_poly(i)
d = zeros([self.order+1,2])
d[0][0] = 1
d[0][1] = 1
sum = 0
for k in range(self.order):
sum += d[k][1]*self.ndd.calc(L, L+k+1)
d[k+1][0] = (self.x[i] - self.x[L+k-1])*d[k][0]
d[k+1][1] = d[k+1][0] + (self.x[i]-self.x[L+k])*d[k][1]
f[i] = sum
for i in range(self.Nb+1, self.N-self.Nb-1):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
class Solver:
def __init__(self):
self.N = 100
self.x = linspace(-2,2,self.N+1)
self.Nb = 10
self.xc = zeros(self.N)
self.u = zeros(self.N)
self.cfl = 0.4
for i in range(self.N):
self.xc[i] = 0.5*(self.x[i] + self.x[i+1])
if abs(self.xc[i]) < 1/3.0:
self.u[i] = 1
self.dx = self.xc[1] - self.xc[0]
self.f = Burgers()
self.order = 3
def calc_poly(self, i, order):
if abs(self.u[i+1]-self.u[i]) > SMALL:
abar = (self.f(self.u[i+1]) - self.f(self.u[i]))/(self.u[i+1]-self.u[i])
else:
abar = 0
if abar >= 0:
k = i
else:
k = i + 1
L = 2
for i in range(order-1):
a = self.ndd.calc(k, k+L)
b = self.ndd.calc(k-1, k+L-1)
if abs(a) >= abs(b):
k = k-1
L += 1
return k
def step(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
for i in range(self.Nb, self.N-self.Nb):
L = self.calc_poly(i)
d = zeros([self.order+1,2])
d[0][0] = 1
d[0][1] = 1
sum = 0
for k in range(self.order):
sum += d[k][1]*self.ndd.calc(L, L+k+1)
d[k+1][0] = (self.x[i] - self.x[L+k-1])*d[k][0]
d[k+1][1] = d[k+1][0] + (self.x[i]-self.x[L+k])*d[k][1]
f[i] = sum
for i in range(self.Nb+1, self.N-self.Nb-1):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def newstep(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
v = zeros_like(self.u)
for i in range(self.Nb+4, self.N-self.Nb-4):
L = self.calc_poly(i, 3)
r = i - L
if r == -1:
v[i] = 11.0/6.0*u_old[i-r+0] - 7.0/6.0*u_old[i-r+1] + 1.0/3.0*u_old[i-r+2]
elif r == 0:
v[i] = 1.0/3.0*u_old[i-r+0] + 5.0/6.0*u_old[i-r+1] - 1.0/6.0*u_old[i-r+2]
elif r == 1:
v[i] = -1.0/6.0*u_old[i-r+0] + 5.0/6.0*u_old[i-r+1] + 1.0/3.0*u_old[i-r+2]
elif r == 2:
v[i] = 1.0/3.0*u_old[i-r+0] - 7.0/6.0*u_old[i-r+1] + 11.0/6.0*u_old[i-r+2]
else:
print "Error"
for i in range(self.Nb+5, self.N-self.Nb-5):
self.alpha = 1.01
#f[i] = 0.5*(self.f(v[i]) + self.f(v[i+1]) - self.alpha*(v[i+1]-v[i]))
f[i] = self.f(v[i])
for i in range(self.Nb+6, self.N-self.Nb-6):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def step4(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
v = zeros_like(self.u)
for i in range(self.Nb+5, self.N-self.Nb-5):
L = self.calc_poly(i, 4)
r = i - L
if r == -1:
v[i] = 25.0/12.0*u_old[i-r+0] - 23.0/12.0*u_old[i-r+1] + 13.0/12.0*u_old[i-r+2] - 1.0/4.0*u_old[i-r+3]
elif r == 0:
v[i] = 1.0/4.0*u_old[i-r+0] + 13.0/12.0*u_old[i-r+1] - 5.0/12.0*u_old[i-r+2] + 1.0/12.0*u_old[i-r+3]
elif r == 1:
v[i] = -1.0/12.0*u_old[i-r+0] + 7.0/12.0*u_old[i-r+1] + 7.0/12.0*u_old[i-r+2] - 1.0/12.0*u_old[i-r+3]
elif r == 2:
v[i] = 1.0/12.0*u_old[i-r+0] - 5.0/12.0*u_old[i-r+1] + 13.0/12.0*u_old[i-r+2] + 1.0/4.0*u_old[i-r+3]
elif r == 3:
v[i] = -1.0/4.0*u_old[i-r+0] + 13.0/12.0*u_old[i-r+1] - 23.0/12.0*u_old[i-r+2] + 25.0/12.0*u_old[i-r+3]
else:
print "Error"
for i in range(self.Nb+6, self.N-self.Nb-6):
self.alpha = 1.01
#f[i] = 0.5*(self.f(v[i]) + self.f(v[i+1]) - self.alpha*(v[i+1]-v[i]))
f[i] = self.f(v[i])
for i in range(self.Nb+7, self.N-self.Nb-7):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def step2(self):
u_old = copy(self.u)
u_new = copy(self.u)
self.ndd = NewtonDD(self.x, u_old, self.f)
f = zeros_like(self.u)
v = zeros_like(self.u)
for i in range(self.Nb+5, self.N-self.Nb-5):
L = self.calc_poly(i, 2)
r = i - L
if r == -1:
v[i] = 3.0/2.0*u_old[i-r+0] - 1.0/2.0*u_old[i-r+1]
elif r == 0:
v[i] = 1.0/2.0*u_old[i-r+0] + 1.0/2.0*u_old[i-r+1]
elif r == 1:
v[i] = -1.0/2.0*u_old[i-r+0] + 3.0/2.0*u_old[i-r+1]
else:
print "Error"
for i in range(self.Nb+6, self.N-self.Nb-6):
self.alpha = 1.01
#f[i] = 0.5*(self.f(v[i]) + self.f(v[i+1]) - self.alpha*(v[i+1]-v[i]))
f[i] = self.f(v[i])
for i in range(self.Nb+7, self.N-self.Nb-7):
self.u[i] -= self.dt/self.dx*(f[i]-f[i-1])
def set_dt(self):
umax = max(abs(self.u))
self.dt = self.cfl*(self.x[1]-self.x[0])/umax
print self.dt
def analytical(self, tf):
shift = tf/self.dx
u = copy(self.u)
for i in range(self.N):
try:
u[i+shift] = self.u[i]
except:
pass
plot(self.xc, u, 'x-')
def solve(self, tf, order):
t = 0
while t < tf:
self.set_dt()
if order == 3:
self.newstep()
elif order == 4:
self.step4()
elif order == 2:
self.step2()
else:
break
t += self.dt