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