def test_diff2fd24():
for n in 2**arange(4,11):
#Create an evenly-spaced grid of n points
x = linspace(-1,1,n)
# Second-order differentiation matrix against second-order derivative
D2 = diff2fd24(n,2,2./(n-1))
D4 = diff2fd24(n,4,2./(n-1))
du_true, u = testf(x,2)
du2 = D2.dot(u)
du4 = D4.dot(u)
# Discrete L2-norm -- right-hand rule
err2 = tools.normd(du2-du_true,x,'even')
print("Discrete L2-error (2nd order) is %20.15e for n = %4d") % (err2,n)
err4 = tools.normd(du4-du_true,x,'even')
print("Discrete L2-error (4th order) is %20.15e for n = %4d") % (err4,n)
def test_diff2lag():
for n in 2**arange(4,7):
#Create Chebyshev points (of the second kind) to interpolate
x = cos(pi*linspace(0,n,n)/n)
# Second-order differentiation matrix against second-order derivative
D = diff2lag(x)
du_true, u = testf(x,2)
du = dot(D,u)
# Discrete L2-norm -- right-hand rule
err = tools.normd(du-du_true,x,'right')
print("Discrete L2-error is %20.15e for n = %4d") % (err,n)
def test():
for n in 2**arange(4,7)-1:
#Create Chebyshev points (of the second kind) to interpolate
x = cos(pi*linspace(0,n,n)/n)
y = testf(x)
# Run the test
ni = 2*(n+1)
xi = linspace(-1+1./n**2,1-1./n**2,ni)
yi = barylag(x,y,xi)
ytrue = testf(xi)
err = tools.normd(yi-ytrue,xi,'right')
print("Discrete L2-error is %20.15e for nodes = %4d, points = %4d") % (err,n,ni)
def test():
t0 = -3
tf = 3
hvec = array([0.1, 0.01, 0.001])
# Euler Test
y0 = sqrt(pi)/2*exp(-t0**2)
print "Scalar Euler Test Results:\n"
for h in hvec:
y, tv = euler(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta = tools.normd(y-ytrue,tv,'even')
print("Discrete L2 error is %20.15e for h = %2.1e") % (delta, h)
y0 = array([1, 2, 3])*sqrt(pi)/2*exp(-t0**2)
delta = zeros(3)
print "\nVector Euler Test Results:\n"
for h in hvec:
y, tv = euler(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta[0] = tools.normd(y[:,0]-ytrue,tv,'even')
delta[1] = tools.normd(y[:,1]-2*ytrue,tv,'even')
delta[2] = tools.normd(y[:,2]-3*ytrue,tv,'even')
print("Discrete L2 error is\n %20.15e, %20.15e, %20.15e\nfor h = %2.1e") % (delta[0],delta[1],delta[2], h)
# Euler Predictor-Corrector Test
y0 = sqrt(pi)/2*exp(-t0**2)
print "Scalar Euler PC Test Results:\n"
for h in hvec:
y, tv = eulerpc(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta = tools.normd(y-ytrue,tv,'even')
print("Discrete L2 error is %20.15e for h = %2.1e") % (delta, h)
y0 = array([1, 2, 3])*sqrt(pi)/2*exp(-t0**2)
delta = zeros(3)
print "\nVector Euler PC Test Results:\n"
for h in hvec:
y, tv = eulerpc(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta[0] = tools.normd(y[:,0]-ytrue,tv,'even')
delta[1] = tools.normd(y[:,1]-2*ytrue,tv,'even')
delta[2] = tools.normd(y[:,2]-3*ytrue,tv,'even')
print("Discrete L2 error is\n %20.15e, %20.15e, %20.15e\nfor h = %2.1e") % (delta[0],delta[1],delta[2], h)
# RK4 Test
y0 = sqrt(pi)/2*exp(-t0**2)
print "Scalar RK4 Test Results:\n"
for h in hvec:
y, tv = RK4(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta = tools.normd(y-ytrue,tv,'even')
print("Discrete L2 error is %20.15e for h = %2.1e") % (delta, h)
y0 = array([1, 2, 3])*sqrt(pi)/2*exp(-t0**2)
delta = zeros(3)
print "\nVector RK4 Test Results:\n"
for h in hvec:
y, tv = RK4(testf,y0,t0,tf,h,True)
ytrue = sqrt(pi)/2*exp(-tv**2)
delta[0] = tools.normd(y[:,0]-ytrue,tv,'even')
delta[1] = tools.normd(y[:,1]-2*ytrue,tv,'even')
delta[2] = tools.normd(y[:,2]-3*ytrue,tv,'even')
print("Discrete L2 error is\n %20.15e, %20.15e, %20.15e\nfor h = %2.1e") % (delta[0],delta[1],delta[2], h)
