def DGreen(N: int, CFL: float):
dx = 2 / N
dt = (CFL * dx**2)
a = -1
b = 1
tf = 0.25
method = "CN"
def d(x):
return 1
def q(x):
return 0.0
def bc0(t):
return 0
def bc1(t):
return 0
def Ddelta(x):
if (x < dx / 4 and x > -dx / 4):
return 1
else:
return 0
def f(x, t):
return 0
BC1 = BoundaryCondition(a, type="N", constraint=lambda t: 0)
BC2 = BoundaryCondition(b, type="N", constraint=lambda t: 0)
# geometry_dict = {"Dirichlet": -1, "Neumann": 0}
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
# xsample = xsample[-geometry_dict[BC1.type]: xsample.shape[0]+geometry_dict[BC2.type]]
# xsample=xsample[1:]
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST,
Nonlinear=None,
IC=Ddelta,
BCs=[BC1, BC2],
rhs=lambda x, t: 0)
ibvp.compute()
# print(ibvp.odefunc(ic(xsample)[0:-1], 0))
sol = ibvp.customize_solve(t_span=[0, tf], type=method, max_step=dt)
sol.t = sol.t[1:]
sol.y = sol.y[:, 1:]
min = sol.y.min()
PlotSpaceTimePDE(
sol,
xsample,
xcount=2,
tcount=2,
title="Discrete Green for {},N={}, CFL={}, min={:.2g}".format(
method, N, CFL, min))
plt.plot(xsample, sol.y[:, 0])
plt.show()
def discretization_error_plot(node_nums=[8, 16, 32, 64, 128, 256, 512, 1024],
a=0,
b=1):
norm_dict = {"L2": 2, "L1": 1, "Linf": np.inf}
mark_dict = {"L2": 'o-', "L1": 's-', "Linf": 'h-'}
for norm_type in norm_dict:
err_list = [
Node_num_rel_err(num, a=0., b=1., norm_type=norm_type)
for num in node_nums
]
plt.loglog(node_nums, err_list, mark_dict[norm_type], label=norm_type)
quadr = lambda x: x**(-2) * 10
plt.loglog(node_nums, [quadr(x) for x in node_nums],
'--',
label="quadratic base line")
plt.xlabel("node numbers")
plt.ylabel("relative error w.r.t. exact function")
plt.legend()
plt.grid(True)
plt.show()
def result(x):
return -3.0 / 8 * (np.pi**2) * (np.cos(np.pi / 2 * x) +
np.cos(3 / 2 * np.pi * x))
# BC1 = BoundaryCondition(a, type="D", constraint=lambda t: 0.)
# BC2 = BoundaryCondition(b, type="N", constraint=lambda t: -0.5 * np.pi)
for num in [2, 4, 5, 6, 8]:
spgrid = SpaceGrid(a, b, num + 1)
xsample = np.linspace(a, b, num + 1)
def d(x):
return 2 + np.cos(np.pi * x)
def u(x):
return np.cos(np.pi * x / 2)
ST = SturmLiouville(spgrid, d)
usample = u(xsample)
ysample = ST(usample)
plt.plot(xsample[1:-1], ysample, label="node num={}".format(num))
plt.plot(np.linspace(a, b, 1000 + 1),
result(np.linspace(a, b, 1000 + 1)),
label="exact")
plt.legend()
plt.grid(True)
plt.xlabel("x")
plt.ylabel("y")
plt.show()
def IBVP_solve_plot():
def d(x):
return 1
def q(x):
return 0
def u(x):
return np.cos(np.pi * x / 2)
def ftilde(xsample, t):
return 0
def ic1(xsample):
return np.array([pow(np.cos(np.pi * 0.5 * x), 100) for x in xsample])
def ic(xsample):
def step(x):
if x < 0.5:
return 0
else:
return 1
return np.array([step(x) for x in xsample])
a = 0
b = 1
BC1 = BoundaryCondition(a, type="D", constraint=lambda t: 0)
BC2 = BoundaryCondition(b, type="N", constraint=lambda t: 0)
geometry_dict = {"Dirichlet": -1, "Neumann": 0}
spgrid = SpaceGrid(a, b, 512 + 1)
xsample = np.linspace(a, b, 512 + 1)
# xsample = xsample[-geometry_dict[BC1.type]: xsample.shape[0]+geometry_dict[BC2.type]]
# xsample=xsample[1:]
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST, Nonlinear=None, IC=ic, BCs=[BC1, BC2], rhs=ftilde)
ibvp.compute()
# print(ibvp.odefunc(ic(xsample)[0:-1], 0))
sol = ibvp.ibvp_solve(t_span=[0, 10], type="Radau", max_step=0.01)
Write_sol_json(sol, file_name="sol.json")
# sol = ibvp.customize_solve(t_span=[0,10], max_step=0.01)
sol = Read_sol_json(file_name="sol.json")
# Animate_1d_wave(sol, xsample=xsample)
PlotSpaceTimePDE(sol, xsample)
def Node_num_rel_err(M=1000, a=0, b=1, norm_type="L1"):
M = M
spgrid = SpaceGrid(a, b, M + 1)
xsample = np.linspace(a, b, M + 1)
def d(x):
return 2 + np.cos(np.pi * x)
def u(x):
return np.cos(np.pi * x / 2)
def result(x):
return -3.0 / 8 * (np.pi**2) * (np.cos(np.pi / 2 * x) +
np.cos(3 / 2 * np.pi * x))
ST = SturmLiouville(spgrid, d)
usample = u(xsample)
ysample = ST(usample)
u_exact = result(xsample)[1:-1]
return function_norm_err(ysample=ysample,
u_exact=u_exact,
norm_type=norm_type)
def BVP_solve_plot(node_nums=[3, 5, 8, 16, 32, 64, 128, 256, 512, 1024]):
mark_dict = dict(
zip([3, 4, 5, 6, 7, 8], ["p-", "o-", "X-", "s-", 'x-', 'D-']))
def d(x):
return 2 + np.cos(np.pi * x)
def q(x):
return 0.0
def u(x):
return np.cos(np.pi * x / 2)
def ftilde(x):
return -3.0 / 8 * (np.pi**2) * (np.cos(np.pi / 2 * x) +
np.cos(3 / 2 * np.pi * x))
a = 0
b = 1
BC1 = BoundaryCondition(a, type="D", constraint=lambda t: 1.)
BC2 = BoundaryCondition(b, type="N", constraint=lambda t: -0.5 * np.pi)
geometry_dict = {"Dirichlet": -1, "Neumann": 0}
norm_errors_L1 = []
norm_errors_L2 = []
norm_errors_Linf = []
dirichlet_errors = []
neumann_errors = []
for num in node_nums:
spgrid = SpaceGrid(a, b, num + 1)
xsample = np.linspace(a, b, num + 1)
# xsample = xsample[-geometry_dict[BC1.type]: xsample.shape[0]+geometry_dict[BC2.type]]
# xsample=xsample[1:]
ST = SturmLiouville(spgrid, d, q)
bvp = BVP(ST=ST, BCs=[BC1, BC2], rhs=ftilde)
bvp.compute()
# print(bvp.solve().shape, xsample.shape)
ysample = bvp.solve()
# print("aha",ST.fdmat.toarray())
# print("ahaaa", ST.p / (ST.h ** 2))
if (num < 10):
plt.plot(xsample,
bvp.solve(),
'X-',
label="node number ={}".format(num))
usample = u(xsample)
norm_errors_L1.append(function_norm_err(ysample, usample, "L1"))
norm_errors_L2.append(function_norm_err(ysample, usample, "L2"))
norm_errors_Linf.append(function_norm_err(ysample, usample, "Linf"))
neumann_errors.append(np.abs(ysample[-1] - usample[-1]))
plt.plot(np.linspace(a, b, 1000),
u(np.linspace(a, b, 1000)),
label="exact")
plt.title("Compare the exact result with small node numbers")
plt.legend()
plt.grid(True)
plt.show()
plt.loglog(node_nums, [2 * error for error in neumann_errors],
'x-',
label="neumann error")
plt.loglog(node_nums, norm_errors_Linf, 's-', label="Linf norm error")
plt.loglog(node_nums, norm_errors_L2, 'o-', label="L2 norm error")
plt.loglog(node_nums, norm_errors_L1, 'p-', label="L1 nodem error")
plt.loglog(node_nums, [pow(num, -2) * 10 for num in node_nums],
'--',
label="second order base line")
plt.legend()
plt.grid(True)
plt.show()
def NonlinearPDE():
a = -1
b = 1
u0 = 26.16
tf = 2.12
dt = 0.01
N = 64
method = "Radau"
def d(x):
return 2 + np.cos(2 * np.pi * x)
def q(x):
return 0
def nonlinear(x):
return x**2
def f(x, t):
return 0
def ic(x):
return u0 * np.cos(np.pi * x / 2)**100
BC1 = BoundaryCondition(a, type="D", constraint=lambda t: 0)
BC2 = BoundaryCondition(b, type="D", constraint=lambda t: 0)
# geometry_dict = {"Dirichlet": -1, "Neumann": 0}
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
# xsample = xsample[-geometry_dict[BC1.type]: xsample.shape[0]+geometry_dict[BC2.type]]
# xsample=xsample[1:]
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST,
Nonlinear=nonlinear,
IC=ic,
BCs=[BC1, BC2],
rhs=lambda x, t: 0)
ibvp.compute()
# print(ibvp.odefunc(ic(xsample)[0:-1], 0))
sol = ibvp.ibvp_solve(t_span=[0, tf], type=method, max_step=dt)
Write_sol_json(sol, file_name="sol.json")
# sol = ibvp.customize_solve(t_span=[0,10], max_step=0.01)
sol = Read_sol_json(file_name="sol.json")
# Animate_1d_wave(sol, xsample=xsample)
sol_atx0 = sol.y[N // 2]
# print("ahah",sol_atx0)
ta = 3 * tf / 4
nta = int(ta // dt)
nta = 0
sol_atx0 = sol_atx0[nta:]
tsample = sol.t[nta:]
print("t", tsample)
print("u", sol_atx0.shape)
T = 2.125
plt.plot(tsample, sol_atx0, 'x', label="f(t)=u(x=0,t)")
plt.plot(tsample, [1 / (T - t) for t in tsample],
label="1/({}-t)".format(T))
plt.legend()
plt.title("finite time blow up and $1/(T-t)$")
plt.grid(True)
plt.xlabel("t")
plt.ylabel("u(x=0, t)")
plt.show()
def CN_BE_Compare(t: float = 0.01, tf: float = 0.25):
def d(x):
return 1
def q(x):
return 0.0
def bc0(t):
return 0
def bc1(t):
return 0
def ic(x):
if x < 0.5:
return 0
else:
return 1
# return np.sin(np.pi * x /2)
def f(x, t):
return 0
a = 0
b = 1
t_span = [0, tf]
N = 128
CFL = 100
dts = [CFL / N**2]
methods = ["BE", "CN"]
markerdict = dict(zip(methods, ["x-", "--"]))
BC0 = BoundaryCondition(a, type="D", constraint=bc0)
BC1 = BoundaryCondition(b, type="N", constraint=bc1)
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
FFT_name = "FFT for tf={}.json".format(tf)
import os
exists = os.path.isfile('../data/{}'.format(FFT_name))
if exists:
# Store configuration file values
print("Read existing FFT solution file")
FFTsol = Read_sol_json(FFT_name)
FFT_xspan = np.linspace(a, b, 1024 + 1)[:-1]
else:
FFTsol, FFT_xspan = FFT_ibvp(t_span,
x_span=[a, b],
Nt=1000,
Nx=512,
ic=ic)
nth = int(t / (tf / 1000))
FFTsol_at_t = FFTsol.y[:, nth]
# initialize the SturmLiouville system and initial-boundary value problem
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST, Nonlinear=None, IC=ic, BCs=[BC0, BC1], rhs=f)
# compute the discretization and matrix involved
ibvp.compute()
sol_at_t_dict = {}
for method in methods:
for dt in dts:
json_name = "sol_compare tspan=({},{}) node={}, dt={:.2g}, for {}.json".format(
0, tf, N, dt, method)
print(json_name)
import os
exists = os.path.isfile('../data/{}'.format(json_name))
if exists:
# Store configuration file values
print("Read existing solution file")
sol = Read_sol_json(json_name)
else:
# Keep presets
if (method in ["CN", "BE"]):
sol = ibvp.customize_solve(t_span=t_span,
type=method,
max_step=dt)
Write_sol_json(sol, json_name)
else:
sol = ibvp.ibvp_solve(t_span=t_span,
type=method,
max_step=dt)
Write_sol_json(sol, json_name)
nth = int(t / dt) + 1
sol_at_t_dict[(method, dt)] = sol.y[:, nth]
if method == "CN":
PlotSpaceTimePDE(
sol,
xsample,
title="Space time sol for CN N={}, CFL={}".format(N, CFL),
xcount=5,
tcount=50)
for key in sol_at_t_dict:
method = key[0]
dt = key[1]
plt.semilogy(xsample[10::2],
np.abs(sol_at_t_dict[key])[10::2],
markerdict[method],
label="solution at t={}, dt={}, CFL={}, method={}".format(
t, dt, N * N * dt, method))
plt.semilogy(FFT_xspan[100:],
FFTsol_at_t[100:],
label="FFT sol at t={}".format(t))
plt.legend(loc='upper center',
bbox_to_anchor=(0.5, -0.2),
fancybox=True,
shadow=True,
ncol=1)
plt.xlabel("x")
plt.ylabel("u(t={})".format(t))
plt.title("Comapre for CFL={} for grid num={}".format(N * N * dt, N))
plt.grid(True)
plt.show()
def Verify2ndAccuracy(grid_size=[4, 8, 16, 32, 64, 128],
axis: str = 't',
norm_type: str = "L1"):
def d(x):
return 2 + np.cos(np.pi * x)
def q(x):
return 0.0
def bc0(t):
return 1 + np.exp(-(np.pi**2) * t / 4)
def bc1(t):
return -0.5 * np.pi * (1 + np.exp(-(np.pi**2) * t / 4))
def ic(x):
return 2 * np.cos(0.5 * np.pi * x)
def f(x, t):
result = -1/ 4 * (np.pi ** 2) * np.exp(- (np.pi ** 2) * t / 4) \
* np.cos(x * np.pi/2)*(-1 + 3 *(1 + np.exp((np.pi ** 2) * t / 4))* np.cos(x * np.pi))
return result
def exact_sol(x, t):
return (1 + np.exp(-(np.pi**2) * t / 4)) * np.cos(np.pi * x / 2)
a = 0
b = 1
t_span = [0, 0.25]
methods = ["CN", "Ralston"]
method = methods[0]
markdict = {4: 'x', 8: 'x', 16: 'x', 32: 'x', 64: 'x', 128: '-', 256: "x"}
markoffsetdict = dict(zip(methods, ['', '-']))
errors2d = {"L1": [], "L2": [], "Linf": []}
for N in grid_size:
dt = 1 / (8 * N**2)
BC0 = BoundaryCondition(a, type="D", constraint=bc0)
BC1 = BoundaryCondition(b, type="N", constraint=bc1)
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
# initialize the SturmLiouville system and initial-boundary value problem
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST, Nonlinear=None, IC=ic, BCs=[BC0, BC1], rhs=f)
# compute the discretization and matrix involved
ibvp.compute()
json_name = "sol node={}, dt={:.2g}, for {}.json".format(N, dt, method)
print(json_name)
import os
exists = os.path.isfile('../data/{}'.format(json_name))
if exists:
# Store configuration file values
sol = Read_sol_json(json_name)
else:
# Keep presets
if (method in ["CN", "BE"]):
sol = ibvp.customize_solve(t_span=t_span,
type=method,
max_step=dt)
else:
sol = ibvp.ibvp_solve(t_span=t_span, type=method, max_step=dt)
Write_sol_json(sol, json_name)
ysample = np.abs(sol.y - np.array([[exact_sol(x, t) for t in sol.t]
for x in xsample]))[1:, 1:]
tsample = sol.t[1:]
sliced_xsample = xsample[1:]
sol1 = OdeResult(t=tsample, y=ysample)
errors = FunctionNormAlong(sol=sol1,
xsample=sliced_xsample,
axis=axis,
type=norm_type)
errors_L1 = FunctionNormAlong(sol=sol1,
xsample=sliced_xsample,
axis=axis,
type="L1")
errors_L2 = FunctionNormAlong(sol=sol1,
xsample=sliced_xsample,
axis=axis,
type="L2")
errors_Linf = FunctionNormAlong(sol=sol1,
xsample=sliced_xsample,
axis=axis,
type="Linf")
errors2d["L1"].append(
FunctionNorm1D(errors_L1.copy(), 1 / N, type="L1"))
errors2d["L2"].append(
FunctionNorm1D(errors_L2.copy(), 1 / N, type="L2"))
errors2d["Linf"].append(
FunctionNorm1D(errors_Linf.copy(), 1 / N, type="Linf"))
if axis == "t":
offset = {4: 1, 8: 1, 16: 1, 32: 1, 64: 2, 128: 4}[N]
plt.semilogy(sliced_xsample[::offset],
errors[::offset],
markdict[N] + markoffsetdict[method],
label=json_name[4:-5])
elif axis == "x":
plt.semilogy(tsample,
errors,
markdict[N] + markoffsetdict[method],
label=json_name[4:-5])
# powers = [pow(2, i) for i in range(1,5)]
for i in [2, 4, 8, 16, 32]:
plt.semilogy(sliced_xsample,
errors * pow(i, 2),
'--',
label="{} times the node=128 error".format(i))
if axis == 't':
plt.xlabel("x")
plt.title(
"Verification of 2nd oder accuracy in {} norm".format(norm_type))
else:
plt.xlabel("t")
plt.title(
"Verification of 2nd oder accuracy in {} norm".format(norm_type))
plt.ylabel("log(error})")
plt.grid(True)
plt.legend(loc='center left', bbox_to_anchor=(1, 0.6), shadow=True, ncol=1)
plt.show()
for key in errors2d:
plt.loglog(grid_size,
errors2d[key],
'x-',
label="total error in 2D-{} norm".format(key))
plt.loglog(grid_size, [4 * pow(g, -2) for g in grid_size],
'-',
label="base line of second oder arruracy")
# plt.loglog(grid_size, [0.01 * pow(g, -4) for g in grid_size], '-', label="base line of fourth oder arruracy")
plt.grid(True)
plt.legend()
plt.xlabel("node number N")
plt.ylabel("total errors")
plt.title(
"Verification of 2nd order accuracy in 2D norms for {}".format(method))
plt.show()
def LTEplot2D(grid_size=[4, 8, 16, 32, 64, 128],
axis: str = 't',
norm_type: str = "L1"):
def d(x):
return 2 + np.cos(np.pi * x)
def q(x):
return 0.0
def bc0(t):
return 1 + np.exp(-(np.pi**2) * t / 4)
def bc1(t):
return -0.5 * np.pi * (1 + np.exp(-(np.pi**2) * t / 4))
def ic(x):
return 2 * np.cos(0.5 * np.pi * x)
def f(x, t):
result = -1/ 4 * (np.pi ** 2) * np.exp(- (np.pi ** 2) * t / 4) \
* np.cos(x * np.pi/2)*(-1 + 3 *(1 + np.exp((np.pi ** 2) * t / 4))* np.cos(x * np.pi))
return result
def exact_sol(x, t):
return (1 + np.exp(-(np.pi**2) * t / 4)) * np.cos(np.pi * x / 2)
a = 0
b = 1
t_span = [0, 0.25]
methods = ["CN", "Ralston"]
markdict = {4: 'x', 8: 'x', 16: 'x', 32: 'x', 64: 'x', 128: 'x', 256: "x"}
markoffsetdict = dict(zip(methods, ['', '']))
for N in grid_size:
dt = 1 / (8 * N**2)
BC0 = BoundaryCondition(a, type="D", constraint=bc0)
BC1 = BoundaryCondition(b, type="N", constraint=bc1)
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
# initialize the SturmLiouville system and initial-boundary value problem
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST, Nonlinear=None, IC=ic, BCs=[BC0, BC1], rhs=f)
# compute the discretization and matrix involved
ibvp.compute()
for method in methods:
json_name = "sol node={}, dt={:.2g}, for {}.json".format(
N, dt, method)
print(json_name)
import os
exists = os.path.isfile('../data/{}'.format(json_name))
if exists:
# Store configuration file values
sol = Read_sol_json(json_name)
else:
# Keep presets
if (method in ["CN", "BE"]):
sol = ibvp.customize_solve(t_span=t_span,
type=method,
max_step=dt)
else:
sol = ibvp.ibvp_solve(t_span=t_span,
type=method,
max_step=dt)
Write_sol_json(sol, json_name)
ysample = np.abs(sol.y -
np.array([[exact_sol(x, t) for t in sol.t]
for x in xsample]))[1:, 1:]
tsample = sol.t[1:]
sliced_xsample = xsample[1:]
sol1 = OdeResult(t=tsample, y=ysample)
errors = FunctionNormAlong(sol=sol1,
xsample=sliced_xsample,
axis=axis,
type=norm_type)
if axis == "t":
plt.semilogy(sliced_xsample,
errors,
markdict[N] + markoffsetdict[method],
label=json_name[4:-5])
elif axis == "x":
plt.semilogy(tsample,
errors,
markdict[N] + markoffsetdict[method],
label=json_name[4:-5])
if axis == 't':
plt.xlabel("x")
plt.title("x" + "-error relation in {} norm".format(norm_type))
else:
plt.xlabel("t")
plt.title("t" + "-error relation in {} norm".format(norm_type))
plt.ylabel("log(error})")
plt.grid(True)
plt.legend(loc='center left', bbox_to_anchor=(1, 0.8), shadow=True, ncol=1)
plt.show()
def LTEplot3D():
def d(x):
return 2 + np.cos(np.pi * x)
def q(x):
return 0.0
def bc0(t):
return 1 + np.exp(-(np.pi**2) * t / 4)
def bc1(t):
return -0.5 * np.pi * (1 + np.exp(-(np.pi**2) * t / 4))
def ic(x):
return 2 * np.cos(0.5 * np.pi * x)
def f(x, t):
result = -1/ 4 * (np.pi ** 2) * np.exp(- (np.pi ** 2) * t / 4) \
* np.cos(x * np.pi/2)*(-1 + 3 *(1 + np.exp((np.pi ** 2) * t / 4))* np.cos(x * np.pi))
return result
def exact_sol(x, t):
return (1 + np.exp(-(np.pi**2) * t / 4)) * np.cos(np.pi * x / 2)
a = 0
b = 1
t_span = [0, 0.25]
N = 10
dt = 0.00213
method = "Ralston"
BC0 = BoundaryCondition(a, type="D", constraint=bc0)
BC1 = BoundaryCondition(b, type="N", constraint=bc1)
spgrid = SpaceGrid(a, b, N + 1)
xsample = np.linspace(a, b, N + 1)
# initialize the SturmLiouville system and initial-boundary value problem
ST = SturmLiouville(spgrid, d, q)
ibvp = IBVP(ST=ST, Nonlinear=None, IC=ic, BCs=[BC0, BC1], rhs=f)
# compute the discretization and matrix involved
ibvp.compute()
if (method in ["CN", "BE"]):
sol = ibvp.customize_solve(t_span=t_span, type=method, max_step=dt)
else:
sol = ibvp.ibvp_solve(t_span=t_span, type=method, max_step=dt)
json_name = "sol_test node={}, dt={}, for {}".format(N, dt, method)
Write_sol_json(sol, json_name + ".json")
sol1 = Read_sol_json(json_name + ".json")
PlotSpaceTimePDE(sol1,
xsample,
title="Space-Time-Solution node={}, dt={}, for {}".format(
N, dt, method),
tcount=50,
xcount=1)
ysample = np.log10(
np.abs(sol1.y - np.array([[exact_sol(x, t) for t in sol1.t]
for x in xsample])))
sol2 = OdeResult(t=sol1.t[1:], y=ysample[1:, 1:])
PlotSpaceTimePDE(sol2,
xsample[1:],
title="Space-Time-Error node={}, dt={} for {}".format(
N, dt, method),
tcount=50,
xcount=1,
error_plot=True)