def XXX(dim, tend, dt, s, h, b, c, d, c_dir="x", d_dir="x", a=1., CN=True):
"""
dim - sparial dimension
s - width of initial profile
h - mesh size
"""
v_c = c / a * getDirection(dim, c_dir)
v_d = d / a * getDirection(dim, d_dir)
v = (v_c + v_d)
E = b / a
if VERBOSITY:
print("=" * 100)
print(
"XX Start test dim = %d , h=%e, b=%e, c=%e, d=%e, c_dir=%s, d_dir=%s, a=%e, s=%e"
% (dim, h, b, c, d, c_dir, d_dir, a, s))
print("=" * 100)
print("initial width s = ", s)
print("diffusion = ", E)
print("total velocity = ", v)
print("tend = ", tend)
print("tolerance = ", TOL)
print("number of elements over s =", s / h)
b0 = sqrt(-log(TAU) * 4 * (s**2 + E * tend))
b1 = sqrt(-log(TAU)) * 2 * s
X0_0 = max(b1, -v[0] * tend + b0)
X0_1 = max(b1, -v[1] * tend + b0)
l_0 = X0_0 + max(v[0] * tend + b0, b1)
l_1 = X0_1 + max(v[1] * tend + b0, b1)
NE_0 = max(int(l_0 / h + 0.5), 1)
NE_1 = max(int(l_1 / h + 0.5), 1)
if dim == 2:
if VERBOSITY:
print("%d x %d grid over %e x %e with element size %e." %
(NE_0, NE_1, l_0, l_1, h))
if NE_0 * NE_1 > NE_MAX:
raise ValueError("too many elements %s." % (NE_0 * NE_1, ))
dom = Rectangle(n0=NE_0, n1=NE_1, l0=l_0, l1=l_1)
x0 = [X0_0, X0_1]
else:
X0_2 = max(b1, -v[2] * tend + b0)
l_2 = X0_2 + max(v[2] * tend + b0, b1)
NE_2 = max(int(l_2 / h + 0.5), 1)
if VERBOSITY:
print(
"%d x %d x %d grid over %e x %e x %e with element size %e." %
(NE_0, NE_1, NE_2, l_0, l_1, l_2, h))
if NE_0 * NE_1 * NE_2 > NE_MAX:
raise ValueError("too many elements %s." % (NE_0 * NE_1 * NE_2, ))
dom = Brick(n0=NE_0, n1=NE_1, ne2=NE_2, l0=l_0, l1=l_1, l2=l_2)
x0 = [X0_0, X0_1, X0_2]
if VERBOSITY:
print("initial location = ", x0)
print("XX", interpolate(uRef(dom, 0., E, s, v, x0),
FunctionOnBoundary(dom)))
fc_BE = TransportPDE(dom, numEquations=1, useBackwardEuler=True)
fc_BE.setValue(M=a, A=-b * kronecker(dom), B=-v_d * a, C=-v_c * a)
fc_BE.getSolverOptions().setVerbosity(VERBOSITY)
fc_BE.getSolverOptions().setTolerance(TOL)
#
fc_BE.getSolverOptions().setPreconditioner(
fc_BE.getSolverOptions().GAUSS_SEIDEL)
fc_BE.getSolverOptions().setNumSweeps(5)
if VERBOSITY: print("Backward Euler Transport created")
fc_CN = TransportPDE(dom, numEquations=1, useBackwardEuler=False)
fc_CN.setValue(M=a, A=-b * kronecker(dom), B=-v_d * a, C=-v_c * a)
fc_CN.getSolverOptions().setVerbosity(VERBOSITY)
fc_CN.getSolverOptions().setTolerance(TOL)
#fc_CN.getSolverOptions().setPreconditioner(fc_CN.getSolverOptions().GAUSS_SEIDEL)
fc_CN.getSolverOptions().setNumSweeps(2)
if VERBOSITY: print("Crank Nicolson Transport created")
dt_CN = fc_CN.getSafeTimeStepSize()
if VERBOSITY: print("time step size by Crank Nicolson=", dt_CN)
U0 = uRef(dom, 0, E, s, v, x0)
U0_e = uRef(dom, 0, E, s, v, x0, True)
fc_CN.setInitialSolution(U0)
fc_BE.setInitialSolution(U0)
initial_error_L2 = sqrt(integrate((U0 - U0_e)**2))
if VERBOSITY:
print("initial Lsup = ", Lsup(U0), Lsup(U0_e))
print("initial integral = ", integrate(U0_e))
print("initial error = ", initial_error_L2)
print("used time step size =", dt)
if not CN:
n = int(ceil(tend / dt))
if VERBOSITY:
print("Solve Backward Euler:")
print("substeps : ", n)
t0 = clock()
for i in range(n):
u = fc_BE.getSolution(dt)
t0 = clock() - t0
else:
if VERBOSITY: print("Solve Crank Nicolson:")
dt = dt_CN
t0 = clock()
u = fc_CN.getSolution(tend)
t0 = clock() - t0
out = QUALITY(u, uRef(dom, tend, E, s, v, x0, True))
print("XX",
interpolate(uRef(dom, tend, E, s, v, x0), FunctionOnBoundary(dom)))
out['time'] = t0
out['tend'] = tend
out['dt'] = dt
out['dx'] = h
if abs(b) > 0:
out["peclet"] = length(v) * s / b
else:
out["peclet"] = 9999999.
# saveVTK("bb.vtu",u0=U0,u_CN=u_CN, uRef=uRef(dom,dt2,E,s,v,X0) )
return out
NE = 50
dom = Rectangle(NE, 1, l1=1. / NE)
dom = Rectangle(NE, NE)
fc = TransportPDE(dom, numEquations=1)
fc.getSolverOptions().setVerbosityOn()
fc.getSolverOptions().setODESolver(SolverOptions.LINEAR_CRANK_NICOLSON)
fc.getSolverOptions().setODESolver(SolverOptions.BACKWARD_EULER)
fc.getSolverOptions().setODESolver(SolverOptions.CRANK_NICOLSON)
fc.setValue(M=1, C=[-1, 0])
x = dom.getX()
u0 = whereNegative(x[0] - 1. / NE)
c = 0
t = 0
saveVTK("u.%s.vtu" % c, u=u0)
fc.setInitialSolution(u0)
dt = fc.getSafeTimeStepSize()
print("u0 =", u0)
T_END = dt
print("dt = ", dt)
while t < T_END:
print("time step t=", t + dt)
u = fc.getSolution(dt)
saveVTK("u.%s.vtu" % (c + 1, ), u=u)
print("u =", u)
c += 1
t += dt
NE=50
dom=Rectangle(NE,1,l1=1./NE)
dom=Rectangle(NE,NE)
fc=TransportPDE(dom,numEquations=1)
fc.getSolverOptions().setVerbosityOn()
fc.getSolverOptions().setODESolver(SolverOptions.LINEAR_CRANK_NICOLSON)
fc.getSolverOptions().setODESolver(SolverOptions.BACKWARD_EULER)
fc.getSolverOptions().setODESolver(SolverOptions.CRANK_NICOLSON)
fc.setValue(M=1,C=[-1,0])
x=dom.getX()
u0=whereNegative(x[0]-1./NE)
c=0
t=0
saveVTK("u.%s.vtu"%c,u=u0)
fc.setInitialSolution(u0)
dt=fc.getSafeTimeStepSize()
print("u0 =",u0)
T_END=dt
print("dt = ",dt)
while t<T_END:
print("time step t=",t+dt)
u=fc.getSolution(dt)
saveVTK("u.%s.vtu"%(c+1,),u=u)
print("u =",u)
c+=1
t+=dt
def XXX(dim,tend,dt, s, h,b,c,d,c_dir="x", d_dir="x", a=1., CN=True):
"""
dim - sparial dimension
s - width of initial profile
h - mesh size
"""
v_c=c/a*getDirection(dim,c_dir)
v_d=d/a*getDirection(dim,d_dir)
v = (v_c+v_d)
E=b/a
if VERBOSITY:
print("="*100)
print("XX Start test dim = %d , h=%e, b=%e, c=%e, d=%e, c_dir=%s, d_dir=%s, a=%e, s=%e"%(dim, h,b,c,d,c_dir, d_dir, a, s))
print("="*100)
print("initial width s = ",s)
print("diffusion = ",E)
print("total velocity = ",v)
print("tend = ", tend)
print("tolerance = ",TOL)
print("number of elements over s =",s/h)
b0=sqrt(- log(TAU) * 4*(s**2+E*tend))
b1=sqrt(- log(TAU)) * 2*s
X0_0=max(b1,-v[0]*tend + b0)
X0_1=max(b1,-v[1]*tend + b0)
l_0=X0_0+max(v[0]*tend + b0 , b1)
l_1=X0_1+max(v[1]*tend + b0 , b1)
NE_0=max(int(l_0/h+0.5),1)
NE_1=max(int(l_1/h+0.5),1)
if dim == 2:
if VERBOSITY: print("%d x %d grid over %e x %e with element size %e."%(NE_0,NE_1,l_0,l_1,h))
if NE_0*NE_1 > NE_MAX:
raise ValueError("too many elements %s."%(NE_0*NE_1,))
dom=Rectangle(n0=NE_0,n1=NE_1,l0=l_0,l1=l_1)
x0=[X0_0, X0_1]
else:
X0_2=max(b1,-v[2]*tend + b0)
l_2=X0_2+max(v[2]*tend + b0 , b1)
NE_2=max(int(l_2/h+0.5),1)
if VERBOSITY: print("%d x %d x %d grid over %e x %e x %e with element size %e."%(NE_0,NE_1,NE_2,l_0,l_1,l_2,h))
if NE_0*NE_1*NE_2 > NE_MAX:
raise ValueError("too many elements %s."%(NE_0*NE_1*NE_2,))
dom=Brick(n0=NE_0,n1=NE_1, ne2=NE_2, l0=l_0,l1=l_1, l2=l_2)
x0=[X0_0, X0_1, X0_2]
if VERBOSITY:
print("initial location = ",x0)
print("XX", interpolate(uRef(dom,0.,E,s,v,x0), FunctionOnBoundary(dom)))
fc_BE=TransportPDE(dom,numEquations=1)
fc_BE.setValue(M=a, A=-b*kronecker(dom), B=-v_d*a, C=-v_c*a)
fc_BE.getSolverOptions().setVerbosity(VERBOSITY)
fc_BE.getSolverOptions().setTolerance(TOL)
#
fc_BE.getSolverOptions().setPreconditioner(fc_BE.getSolverOptions().GAUSS_SEIDEL)
fc_BE.getSolverOptions().setNumSweeps(5)
if VERBOSITY: print("Backward Euler Transport created")
fc_CN=TransportPDE(dom,numEquations=1)
fc_CN.setValue(M=a, A=-b*kronecker(dom), B=-v_d*a, C=-v_c*a)
fc_CN.getSolverOptions().setVerbosity(VERBOSITY)
fc_CN.getSolverOptions().setTolerance(TOL)
#fc_CN.getSolverOptions().setPreconditioner(fc_CN.getSolverOptions().GAUSS_SEIDEL)
fc_CN.getSolverOptions().setNumSweeps(2)
if VERBOSITY: print("Crank Nicolson Transport created")
dt_CN=fc_CN.getSafeTimeStepSize()
if VERBOSITY: print("time step size by Crank Nicolson=",dt_CN)
U0=uRef(dom,0,E,s,v,x0)
U0_e=uRef(dom,0,E,s,v,x0,True)
fc_CN.setInitialSolution(U0)
fc_BE.setInitialSolution(U0)
initial_error_L2=sqrt(integrate((U0-U0_e)**2))
if VERBOSITY:
print("initial Lsup = ",Lsup(U0), Lsup(U0_e))
print("initial integral = ",integrate(U0_e))
print("initial error = ",initial_error_L2)
print("used time step size =",dt)
if not CN:
n=int(ceil(tend/dt))
if VERBOSITY:
print("Solve Backward Euler:")
print("substeps : ",n)
t0=clock()
for i in range(n): u=fc_BE.getSolution(dt)
t0=clock()-t0
else:
if VERBOSITY: print("Solve Crank Nicolson:")
dt=dt_CN
t0=clock()
u=fc_CN.getSolution(tend)
t0=clock()-t0
out=QUALITY(u,uRef(dom,tend,E,s,v,x0,True))
print("XX", interpolate(uRef(dom,tend,E,s,v,x0), FunctionOnBoundary(dom)))
out['time']=t0
out['tend']=tend
out['dt']=dt
out['dx']=h
if abs(b)>0:
out["peclet"]=length(v)*s/b
else:
out["peclet"]=9999999.
# saveVTK("bb.vtu",u0=U0,u_CN=u_CN, uRef=uRef(dom,dt2,E,s,v,X0) )
return out