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
# set initial value x = dom.getX() u0 = 1 / (4. * pi * E * T0)**(DIM / 2.) * exp( -length(dom.getX() - getCenter(T0))**2 / (4. * E * T0)) print("QUALITY ", QUALITY(T0, u0)) x = Function(dom).getX() if DIM == 2: V = OMEGA0 * (x[0] * [0, -1] + x[1] * [1, 0]) else: V = OMEGA0 * (x[0] * [0, cos(ALPHA), 0] + x[1] * [-cos(ALPHA), 0, sin(ALPHA)] + x[2] * [0., -sin(ALPHA), 0.]) #=================== fc = TransportPDE(dom, num_equations=1, theta=THETA) x = Function(dom).getX() fc.setValue(M=Scalar(1., Function(dom)), C=V, A=-Scalar(E, Function(dom)) * kronecker(dom)) #============== if TEST_SUPG: supg = LinearSinglePDE(dom) supg.setValue(D=1.) supg.setSolverMethod(supg.LUMPING) dt_supg = 1. / (1. / inf(dom.getSize() / length(V)) + 1. / inf(dom.getSize()**2 / E)) * 0.3 u_supg = u0 * 1. c = 0 saveVTK("u.%s.vtu" % c, u=u0)
Primary Business: Queensland, Australia""" __license__ = """Licensed under the Apache License, version 2.0 http://www.apache.org/licenses/LICENSE-2.0""" __url__ = "https://launchpad.net/escript-finley" from esys.escript import * from esys.escript.linearPDEs import TransportPDE, SolverOptions from esys.finley import Rectangle, Brick #from esys.ripley import Rectangle, Brick from esys.weipa import saveVTK from math import pi, ceil 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()