def direct_problem(a, b, eps, M, N, u_left, u_right, t, x, q, h, tau,
u_init):
def func(y, t, x, q):
f = np.zeros(N - 1)
f[0] = (eps * (y[1] - 2 * y[0] + u_left) /
h**2) + (y[0] * (y[1] - u_left) / (2 * h)) - y[0] * q[1]
for n in range(1, N - 2):
f[n] = (eps * (y[n + 1] - 2 * y[n] + y[n - 1]) /
h**2) + (y[n] * (y[n + 1] - y[n - 1]) /
(2 * h)) - y[n] * q[n + 1]
f[N - 2] = (eps * (u_right - 2 * y[N - 2] + y[N - 3]) /
h**2) + (y[N - 2] * (u_right - y[N - 3]) /
(2 * h)) - y[N - 2] * q[N - 1]
return f
u = np.zeros((M + 1, N + 1))
y = np.zeros((M + 1, N - 1))
for n in range(N + 1):
u[0, n] = u_init[n]
y[0, :] = u[0, 1:N]
for m in range(M):
a_diag, b_diag, c_diag = Utils.DiagonalPreparationDirect(
N, eps, tau, q, h, y[m, :], u_left, u_right)
w_1 = Utils.TridiagonalMatrixAlgorithm(
a_diag, b_diag, c_diag,
func(y[m, :], (t[m] + t[m + 1]) / 2, x, q))
tmp2 = tau * w_1.real
y[m + 1] = y[m] + np.transpose(tmp2)
u[m + 1, 1:N] = y[m + 1]
u[m + 1, 0] = u_left
u[m + 1, N] = u_right
return u
def direct_problem(eps, M, N, u_left, u_right, t, x, q, h):
def q_init(x):
return 2 * x - 1 + 2 * np.sin(5 * x * np.pi) + 0.35
for n in range(0, N + 1):
init_q.append(q_init(x[n]))
def u_init(x):
return (x**2 - x - 2) - 6 * np.tanh((-3 * x + 0.75) / eps)
def func(y, t, x, q):
f = np.zeros(N - 1)
f[0] = (eps *
(y[1] - 2 * y[0] + u_left) / h**2) + (y[0] * (y[1] - u_left) /
(2 * h)) - y[0] * q[1]
for n in range(1, N - 2):
f[n] = (eps * (y[n + 1] - 2 * y[n] + y[n - 1]) / h ** 2) \
+ (y[n] * (y[n + 1] - y[n - 1]) / (2 * h)) \
- y[n] * q[n + 1]
f[N - 2] = (eps * (u_right - 2 * y[N - 2] + y[N - 3]) / h ** 2) \
+ (y[N - 2] * (u_right - y[N - 3]) / (2 * h)) \
- y[N - 2] * q[N - 1]
return f
u = np.zeros((M + 1, N + 1))
y = np.zeros((M + 1, N - 1))
for n in range(N + 1):
u[0, n] = u_init(x[n])
y[0, :] = u[0, 1:N]
for m in range(M):
a_diag, b_diag, c_diag = Utils.DiagonalPreparationDirect(
N, eps, tau, q, h, y[m, :], u_left, u_right)
w_1 = Utils.TridiagonalMatrixAlgorithm(
a_diag, b_diag, c_diag, func(y[m, :], (t[m] + t[m + 1]) / 2, x, q))
tmp2 = tau * w_1.real
y[m + 1] = y[m] + np.transpose(tmp2)
u[m + 1, 1:N] = y[m + 1]
u[m + 1, 0] = u_left
u[m + 1, N] = u_right
return u
