def main(gui):
# Constants
l = 10 # domain length
a = 3. # advection velocity
n = 3 # number of elements
p = 4 # order of discretization
v = ['u', 'v'] # physical variables
cfl = 1 / (2*p+1) # max. Courant-Friedrichs-Levy for stability
# Functions
def initial(x, t): return 0.0
def funa(x, t): return np.sin(2*np.pi*(x-a*t)/l*2)
def funb(x, t): return np.sin(2*np.pi*(x+a*t)/l*2)
funs = [funa, funb]
if gui:
gui.vars = v
gui.frefs = funs
# Parameters
dx = l / n # cell length
dt = cfl * dx / a # time step
tmax = round(l / a, 5) # simulation time (l/a)
# Generate mesh
msh = lmsh.run(l, n)
fld = msh.groups[0] # field
inl = msh.groups[1] # inlet
oul = msh.groups[2] # outlet
# Generate formulation
pflx = pfl.Advection2(a, -a) # physical transport flux
ic = numc.Initial(fld, [initial, initial]) # initial condition
inlet = numc.Boundary(inl, [numc.Dirichlet(funa), numc.Neumann()]) # inlet bc
outlet = numc.Boundary(oul, [numc.Neumann(), numc.Dirichlet(funb)]) # outlet bc
formul = numf.Formulation(msh, fld, len(v), pflx, ic, [inlet, outlet])
# Generate discretization
nflx = nfl.LaxFried(pflx, 0.) # Lax–Friedrichs flux (0: full-upwind, 1: central)
disc = numd.Discretization(formul, p, nflx)
# Define time integration method
wrt = wrtr.Writer('sol', 1, v, disc)
tint = numt.SspRk4(disc, wrt, gui)
tint.run(dt, tmax)
# Test
uexact = [[] for _ in range(len(v))] # exact solution at element eval point
ucomp = [[] for _ in range(len(v))] # computed solution at element eval point
for c,e in disc.elements.items():
xe = e.evalx()
for j in range(len(v)):
for i in range(len(xe)):
uexact[j].append(funs[j](xe[i], tint.t))
ucomp[j].append(tint.u[e.rows[j][i]])
maxdiff = [] # infinite norm
nrmdiff = [] # Frobenius norm
for j in range(len(v)):
maxdiff.append(np.max(np.abs(np.array(ucomp[j]) - np.array(uexact[j]))))
nrmdiff.append(np.linalg.norm(np.array(ucomp[j]) - np.array(uexact[j])))
tests = tst.Tests()
tests.add(tst.Test('Max(u-u_exact)', maxdiff[0], 0., 3e-1))
tests.add(tst.Test('Norm(u-u_exact)', nrmdiff[0], 0., 3e-1))
tests.add(tst.Test('Max(v-v_exact)', maxdiff[1], 0., 3e-1))
tests.add(tst.Test('Norm(v-v_exact)', nrmdiff[1], 0., 3e-1))
tests.run()
def main(gui):
# Constants
l = 400 # domain length
g = 9.81 # acceleration due to gravity
h = [1.0, 0.01] # steady state water height
w = [0.1, 20] # height and width of initial wave
z = [20, 100] # bed slope initial and final position
n = 50 # number of elements
p = 6 # order of discretization
v = ['h', 'u'] # physical variables
cfl = 1 / (2*p+1) # half of max. Courant-Friedrichs-Levy for stability
# Functions
# bed
def zb(x):
if x < l/2 + z[0]: return 0.
elif x < l/2 + z[1]: return 0.5 * (h[0]-h[1]) * (1 - np.cos(np.pi/(z[1]-z[0]) * (x-l/2-z[0])))
else: return h[0] - h[1]
# source term function (bed slope * g)
def gdzb(x):
if x >= l/2 + z[0] and x < l/2 + z[1]: return g * 0.5 * np.pi * (h[0]-h[1])/(z[1]-z[0]) * np.sin(np.pi/(z[1]-z[0])*(x-l/2-z[0]))
else: return 0.
# initial and boundary conditions
def h0(x, t): return h[0] + w[0] * np.cos(0.5*np.pi/w[1]*(x-l/2)) - zb(x) if abs(x-l/2) < w[1] else h[0] - zb(x)
def u0(x, t): return 0.0
def hl(x, t): return h[0] - zb(x)
def ul(x, t): return 0.0
# reference solutions
def rfunh(x, t):
if x < 116:
return h[0]
elif x < 160:
return -0.0001*x*x + 0.0272*x - 0.7985
elif x < 220:
return h[0]
elif x < 272:
return -0.000192559021192277*x*x + 0.0810028508376352*x - 7.48640208379703
elif x < 300:
return 0.000308903323667424*x*x - 0.186483075379383*x + 28.1505055888591
else:
return h[1]
def rfunu(x, t):
if x < 116:
return 0.
elif x < 156:
return 0.000353056759525554*x*x - 0.0955691190357186*x + 6.32071941182482
elif x < 245:
return -1.12067281188971e-06*x*x + 0.000289440705269957*x - 0.0167770084481066
elif x < 264:
return -3.37962520178778e-05*x*x + 0.0300418744173713*x - 5.31085043131794
elif x < 272:
return -0.00755364567412781*x*x + 4.01701626023094*x - 533.785818827200
else:
return 0.
funs = [rfunh, rfunu]
if gui:
gui.vars = v
gui.frefs = funs
# Parameters
dx = l / n # cell length
dt = cfl * dx / (0.5 + np.sqrt(g*h[0])) # time step
tmax = 20 # simulation time (l/a)
# Generate mesh
msh = lmsh.run(l, n)
fld = msh.groups[0] # field
inl = msh.groups[1] # inlet
oul = msh.groups[2] # outlet
# Generate formulation
pflx = pfl.ShallowWater(g) # physical transport flux
ic = numc.Initial(fld, [h0, u0]) # initial condition
left = numc.Boundary(inl, [numc.Dirichlet(hl), numc.Dirichlet(ul)]) # left bc
right = numc.Boundary(oul, [numc.Dirichlet(hl), numc.Dirichlet(ul)]) # right bc
formul = numf.Formulation(msh, fld, len(v), pflx, ic, [left, right], nsrc.Source([lambda x: 0., gdzb]))
# Generate discretization
nflx = nfl.LaxFried(pflx, 0.) # Lax–Friedrichs flux (0: full-upwind, 1: central)
disc = numd.Discretization(formul, p, nflx)
# Define time integration method
wrt = wrtr.Writer('sol', 1, v, disc)
tint = numt.SspRk4(disc, wrt, gui)
tint.run(dt, tmax)
# Test
uexact = [[] for _ in range(len(v))] # exact solution at element eval point
ucomp = [[] for _ in range(len(v))] # computed solution at element eval point
for c,e in disc.elements.items():
xe = e.evalx()
for j in range(len(v)):
for i in range(len(xe)):
uexact[j].append(funs[j](xe[i], tint.t))
ucomp[j].append(tint.u[e.rows[j][i]])
maxdiff = [] # infinite norm
nrmdiff = [] # Frobenius norm
for j in range(len(v)):
maxdiff.append(np.max(np.abs(np.array(ucomp[j]) - np.array(uexact[j]))))
nrmdiff.append(np.linalg.norm(np.array(ucomp[j]) - np.array(uexact[j])))
tests = tst.Tests()
tests.add(tst.Test('Max(h-h_exact)', maxdiff[0], 0., 1e-1))
tests.add(tst.Test('Norm(h-h_exact)', nrmdiff[0], 0., 1e-1))
tests.add(tst.Test('Max(u-u_exact)', maxdiff[1], 0., 1e-1))
tests.add(tst.Test('Norm(u-u_exact)', nrmdiff[1], 0., 1e-1))
tests.run()
def main(gui):
# Constants
l = 1 # domain length
n = 75 # number of elements
p = 2 # order of discretization
gamma = 1.4 # heat capacity ratio
v = ['rho', 'rhou', 'E'] # physical variables
cfl = 1 / (2 * p + 1) # max. Courant-Friedrichs-Levy for stability
# Functions
def fun0(x, t):
return 1. if x < l / 2 else 0.125
def fun1(x, t):
return 0.
def fun2(x, t):
return 1. / (gamma - 1) if x < l / 2 else 0.1 / (gamma - 1)
def rfun0(x, t):
if x < 0.381 * l:
return 1.
elif x < 0.489 * l:
return 15.5391 * x * x - 18.7087 * x + 5.8748
elif x < 0.587 * l:
return 0.442
elif x < 0.665 * l:
return 0.274
else:
return 0.125
def rfun1(x, t):
if x < 0.381 * l:
return 0.
elif x < 0.489 * l:
return -36.1960 * x * x + 35.0524 * x - 8.0979
elif x < 0.587 * l:
return 0.394
elif x < 0.665 * l:
return 0.244
else:
return 0.
def rfun2(x, t):
if x < 0.381 * l:
return 2.5
elif x < 0.489 * l:
return 50.3024 * x * x - 58.8475 * x + 17.6300
elif x < 0.587 * l:
return 0.885
elif x < 0.665 * l:
return 0.831
else:
return 0.25
funs = [rfun0, rfun1, rfun2]
if gui:
gui.vars = v
gui.frefs = funs
# Parameters
dx = l / n # cell length
dt = cfl * dx / 1.0 # time step
tmax = 0.1 # simulation time
# Generate mesh and get groups
msh = lmsh.run(l, n)
fld = msh.groups[0] # field
inl = msh.groups[1] # inlet
oul = msh.groups[2] # outlet
# Generate formulation
pflx = pfl.Euler(gamma) # physical Euler flux
ic = numc.Initial(fld, [fun0, fun1, fun2]) # initial condition
dbcs = [numc.Dirichlet(fun0),
numc.Dirichlet(fun1),
numc.Dirichlet(fun2)] # Dirichlet BCs
bcs = [numc.Boundary(inl, dbcs),
numc.Boundary(oul, dbcs)] # inlet-outlet bc
formul = numf.Formulation(msh, fld, len(v), pflx, ic, bcs)
# Generate discretization
nflx = nfl.LaxFried(pflx,
0.) # Lax–Friedrichs flux (0: full-upwind, 1: central)
disc = numd.Discretization(formul, p, nflx)
# Define time integration method
wrt = wrtr.Writer('sol', 1, v, disc)
tint = numt.SspRk4(disc, wrt, gui)
tint.run(dt, tmax)
# Test
uexact = [[]
for _ in range(len(v))] # exact solution at element eval point
ucomp = [[]
for _ in range(len(v))] # computed solution at element eval point
for c, e in disc.elements.items():
xe = e.evalx()
for j in range(len(v)):
for i in range(len(xe)):
uexact[j].append(funs[j](xe[i], tint.t))
ucomp[j].append(tint.u[e.rows[j][i]])
maxdiff = [] # infinite norm
nrmdiff = [] # Frobenius norm
for j in range(len(v)):
maxdiff.append(np.max(np.abs(np.array(ucomp[j]) -
np.array(uexact[j]))))
nrmdiff.append(
np.linalg.norm(np.array(ucomp[j]) - np.array(uexact[j])))
tests = tst.Tests()
tests.add(tst.Test('Max(rho-rho_exact)', maxdiff[0], 0., 2e-1))
tests.add(tst.Test('Norm(rho-rho_exact)', nrmdiff[0], 0., 5e-1))
tests.add(tst.Test('Max(rhou-rhou_exact)', maxdiff[1], 0., 4e-1))
tests.add(tst.Test('Norm(rhou-rhou_exact)', nrmdiff[1], 0., 5e-1))
tests.add(tst.Test('Max(E-E_exact)', maxdiff[2], 0., 9e-1))
tests.add(tst.Test('Norm(E-E_exact)', nrmdiff[2], 0., 2e-0))
tests.run()
def main(gui):
# Constants
l = 10 # domain length
n = 3 # number of elements
p = 5 # order of discretization
u1 = 1.0 # in-out velocity
v = ['u'] # physical variables
cfl = 0.5 * 1 / (2 * p + 1
) # half of max. Courant-Friedrichs-Levy for stability
# Functions
def initial(x, t):
return -u1 * np.sign(x - l / 2) if np.abs(
x - l / 2) > l / n else -n * u1 / l * (x - l / 2)
def inout(x, t):
return -u1 * np.sign(x - l / 2)
def fun(x, t):
return -u1 * np.sign(x - l / 2)
if gui:
gui.vars = v
gui.frefs = [fun]
# Parameters
dx = l / n # cell length
dt = cfl * dx / abs(u1) # time step
tmax = 5.0 # simulation time
# Generate mesh and get groups
msh = lmsh.run(l, n)
fld = msh.groups[0] # field
inl = msh.groups[1] # inlet
oul = msh.groups[2] # outlet
# Generate formulation
pflx = pfl.Burger() # physical Burger's flux
ic = numc.Initial(fld, [initial]) # initial condition
dbc = [numc.Dirichlet(inout)] # Dirichlet BC
bcs = [numc.Boundary(inl, dbc), numc.Boundary(oul, dbc)] # inlet-outlet bc
formul = numf.Formulation(msh, fld, len(v), pflx, ic, bcs)
# Generate discretization
nflx = nfl.LaxFried(pflx,
0.) # Lax–Friedrichs flux (0: full-upwind, 1: central)
disc = numd.Discretization(formul, p, nflx)
# Define time integration method
wrt = wrtr.Writer('sol', 1, v, disc)
tint = numt.Rk4(disc, wrt, gui)
tint.run(dt, tmax)
# Test
uexact = [] # exact solution at element eval point
for c, e in disc.elements.items():
xe = e.evalx()
for i in range(len(xe)):
ue = fun(xe[i], tint.t)
uexact.append(ue)
maxdiff = np.max(np.abs(tint.u - np.array(uexact))) # infinite norm
nrmdiff = np.linalg.norm(tint.u - np.array(uexact)) # Frobenius norm
tests = tst.Tests()
tests.add(tst.Test('Max(u-u_exact)', maxdiff, 0., 1e-1))
tests.add(tst.Test('Norm(u-u_exact)', nrmdiff, 0., 1e-1))
tests.run()