def calc_solution(courant, dx, axes, plot_initial=False):
xmax = 4.0
tmax = 1.0
speed = 2.0
dt = dx * courant / speed
nx = int(xmax / dx)
nt = int(tmax / dt)
x = np.linspace(0.0, xmax, nx)
u = np.ones(nx)
u[int(0.5 / dx):int(1.0 / dx + 1)] = speed
strings = []
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dt = {dt:>5.2e}")
strings.append(f"CFL = {speed * dt / dx:>5.2e}")
print(" | ".join(strings))
if plot_initial:
axes.plot(x, u, "r-", label="Initial Solution")
for _ in range(nt):
solver(dx, dt, u)
axes.title.set_text(f"Time = {dt*nt:12.5f}")
axes.plot(x, u, "-", label=f"Courant = {courant:.3f}")
def calc_solution(diffusion, nu, dx, axes, plot_initial=False):
xmax = 4.0
tmax = 1.0
speed = 2.0
dt = diffusion * dx * dx / nu
nx = int(xmax / dx)
nt = int(tmax / dt) + 1
x = np.linspace(0.0, xmax, nx)
u = np.ones(nx)
u[int(0.5 / dx):int(1.0 / dx + 1)] = speed
strings = []
strings.append(f"nu = {nu:>5.2e}")
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dt = {dt:>5.2e}")
strings.append(f"CFL = {speed * dt / dx:>5.2e}")
strings.append(f"VIS = {dt * nu / dx / dx:>5.2e}")
print(" | ".join(strings))
if plot_initial:
axes.plot(x, u, "r-", label="Initial Solution")
for _ in range(nt):
solver(nu, dx, dt, u)
axes.title.set_text(f"Time = {dt*nt:12.5f}")
axes.plot(x, u, "-", label=f"Diffusion = {diffusion:.3f} (nu = {nu:.2e})")
def calc_solution(diffusion, nu, dx, axes, plot_initial=False):
xmax = 2.0 * np.pi
tmax = 0.1
dt = diffusion * dx * dx / nu
nx = int(xmax / dx)
nt = int(tmax / dt) + 1
func = get_function()
x = np.linspace(0.0, xmax, nx)
u = np.asarray([func(0.0, x0, nu) for x0 in x])
strings = []
strings.append(f"nu = {nu:>5.2e}")
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dt = {dt:>5.2e}")
strings.append(f"CFL = {max(u) * dt / dx:>5.2e}")
strings.append(f"VIS = {dt * nu / dx / dx:>5.2e}")
print(" | ".join(strings))
if plot_initial:
axes.plot(x, u, "r-", label="Initial Solution")
for _ in range(nt):
solver(nu, dx, dt, u)
axes.title.set_text(f"Time = {dt*nt:12.5f}")
axes.plot(x, u, "-", label=f"Diffusion = {diffusion:.3f} (nu = {nu:.2e})")
def calc_solution(inner, diffusion, nu, dx, dy):
xmax = 5.0
ymax = 2.0
tmax = 5.0
rho = 1.0
dt = diffusion * min(dx, dy)**2 / nu
nx = int(xmax / dx)
ny = int(ymax / dy)
nt = int(tmax / dt) + 1
x = np.linspace(0.0, xmax, nx)
y = np.linspace(0.0, ymax, ny)
X, Y = np.meshgrid(x, y, indexing="ij")
u = np.zeros((nx, ny), order="F")
v = np.zeros((nx, ny), order="F")
p = np.ones((nx, ny), order="F")
f = np.ones((nx, ny), order="F") * 5.0
title = "Navier-Stokes Channel Flow (2D)"
print("-".center(80, "-"))
print(title.center(80))
print("-".center(80, "-"))
strings = []
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dy = {dy:>5.2e}")
strings.append(f"dt = {dt:>5.2e}")
strings.append(f"CFL = {1.0 * dt / min(dx, dy):>5.2e}")
strings.append(f"VIS = {dt * nu / min(dx, dy)**2:>5.2e}")
print(" | ".join(strings))
for _ in range(nt):
solver(inner, dx, dy, dt, rho, nu, f, p, u, v)
figure = plt.figure(figsize=(15, 8), dpi=100)
figure.suptitle(title)
colormap = cm.viridis # @UndefinedVariable
axes = figure.add_subplot(111)
axes.title.set_text("Velocity U")
k = 2
contours = axes.contourf(X, Y, u, 10, alpha=0.5)
axes.contour(X, Y, u, cmap=colormap)
axes.quiver(X[::k, ::k], Y[::k, ::k], u[::k, ::k], v[::k, ::k])
divider = make_axes_locatable(axes)
cax = divider.append_axes("right", size="5%", pad=0.05)
figure.colorbar(contours, cax=cax, orientation="vertical")
axes.set_xlabel("x")
axes.set_ylabel("y")
axes.set_aspect("equal")
plt.tight_layout()
def calc_solution(iterations, dx, dy, plot_3d):
xmax = 2.0
ymax = 1.0
nx = int(xmax / dx)
ny = int(ymax / dy)
x = np.linspace(0.0, xmax, nx)
y = np.linspace(0.0, ymax, ny)
X, Y = np.meshgrid(x, y, indexing="ij")
p = np.zeros((nx, ny), order="F")
p[-1, :] = y
title = "Laplace Equation (2D)"
figure = plt.figure(figsize=(11, 7), dpi=100)
figure.suptitle(title)
colormap = cm.viridis # @UndefinedVariable
axes = {}
if plot_3d:
axes[0] = figure.add_subplot(121, projection="3d")
axes[1] = figure.add_subplot(122, projection="3d")
for i in [0, 1]:
axes[i].view_init(30, 225)
axes[i].set_xlabel("x")
axes[i].set_ylabel("y")
axes[i].set_zlabel("u")
axes[0].plot_surface(X, Y, p, cmap=colormap, rstride=1, cstride=1)
else:
axes[0] = figure.add_subplot(121)
axes[1] = figure.add_subplot(122)
for i in [0, 1]:
axes[i].set_xlabel("x")
axes[i].set_ylabel("y")
contours = axes[0].contourf(X, Y, p)
axes[0].set_aspect("equal")
divider = make_axes_locatable(axes[0])
cax = divider.append_axes("right", size="5%", pad=0.05)
figure.colorbar(contours, cax=cax, orientation="vertical")
axes[0].title.set_text("Initial Solution")
axes[1].title.set_text("Final Solution")
for _ in range(iterations):
residual = solver(dx, dy, p)
print("-".center(80, "-"))
print(title.center(80))
print("-".center(80, "-"))
strings = []
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dy = {dy:>5.2e}")
strings.append(f"iterations = {iterations:>5d}")
strings.append(f"residual = {residual:>5.2e}")
print(" | ".join(strings))
if plot_3d:
axes[1].plot_surface(X, Y, p, cmap=colormap, rstride=1, cstride=1)
else:
contours = axes[1].contourf(X, Y, p)
axes[1].set_aspect("equal")
divider = make_axes_locatable(axes[1])
cax = divider.append_axes("right", size="5%", pad=0.05)
figure.colorbar(contours, cax=cax, orientation="vertical")
plt.tight_layout()
def calc_solution(courant, dx, dy, plot_3d=False):
xmax = 2.0
ymax = 2.0
tmax = 0.5
speed = 2.0
dt = min(dx, dy) * courant / speed
nx = int(xmax / dx)
ny = int(ymax / dy)
nt = int(tmax / dt)
x = np.linspace(0.0, xmax, nx)
y = np.linspace(0.0, ymax, ny)
X, Y = np.meshgrid(x, y, indexing="ij")
u = np.ones((nx, ny), order="F")
for j in range(ny):
for i in range(nx):
if (0.5 <= X[i, j] <= 1.0) and (0.5 <= Y[i, j] <= 1.0):
u[i, j] = speed
title = "Linear Convection (2D)"
print("-".center(80, "-"))
print(title.center(80))
print("-".center(80, "-"))
strings = []
strings.append(f"dx = {dx:>5.2e}")
strings.append(f"dy = {dy:>5.2e}")
strings.append(f"dt = {dt:>5.2e}")
strings.append(f"CFL = {speed * dt / min(dx, dy):>5.2e}")
print(" | ".join(strings))
figure = plt.figure(figsize=(11, 7), dpi=100)
figure.suptitle(title)
colormap = cm.viridis # @UndefinedVariable
axes = {}
if plot_3d:
axes[0] = figure.add_subplot(121, projection="3d")
axes[1] = figure.add_subplot(122, projection="3d")
for i in [0, 1]:
axes[i].view_init(30, 225)
axes[i].set_xlabel("x")
axes[i].set_ylabel("y")
axes[i].set_zlabel("u")
axes[0].plot_surface(X, Y, u, cmap=colormap, rstride=1, cstride=1)
else:
axes[0] = figure.add_subplot(121)
axes[1] = figure.add_subplot(122)
for i in [0, 1]:
axes[i].set_xlabel("x")
axes[i].set_ylabel("y")
contours = axes[0].contourf(X, Y, u)
axes[0].set_aspect("equal")
divider = make_axes_locatable(axes[0])
cax = divider.append_axes("right", size="5%", pad=0.05)
figure.colorbar(contours, cax=cax, orientation="vertical")
axes[0].title.set_text("Initial Solution")
axes[1].title.set_text("Final Solution")
for _ in range(nt):
solver(dx, dy, dt, u)
if plot_3d:
axes[1].plot_surface(X, Y, u, cmap=colormap, rstride=1, cstride=1)
else:
contours = axes[1].contourf(X, Y, u)
axes[1].set_aspect("equal")
divider = make_axes_locatable(axes[1])
cax = divider.append_axes("right", size="5%", pad=0.05)
figure.colorbar(contours, cax=cax, orientation="vertical")
plt.tight_layout()
