#!/usr/bin/env python # -*- coding: utf-8 -*- """ *********************************************************************************** tutorial_cv_8.py DAE Tools: pyDAE module, www.daetools.com Copyright (C) Dragan Nikolic *********************************************************************************** DAE Tools is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License version 3 as published by the Free Software Foundation. DAE Tools is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with the DAE Tools software; if not, see <http://www.gnu.org/licenses/>. ************************************************************************************ """ __doc__ = """ Code verification using the Method of Manufactured Solutions. Reference (page 64): - W. Hundsdorfer. Numerical Solution of Advection-Diffusion-Reaction Equations. Lecture notes (2000), Thomas Stieltjes Institute. `PDF <http://homepages.cwi.nl/~willem/Coll_AdvDiffReac/notes.pdf>`_ The problem in this tutorial is 1D *transient convection-reaction* equation: .. code-block:: none dc/dt + dc/dx = c**2 The exact solution is: .. code-block:: none c(x,t) = sin(pi*(x-t))**2 / (1 - t*sin(pi*(x-t))**2) The equation is solved using the high resolution cell-centered finite volume upwind scheme with flux limiter described in the article. The boundary and initial conditions are obtained from the exact solution. The consistent discretisation of the convection and the source terms cannot be done since the constant C1 in the integral of the source term: .. code-block:: none S(x) = 1/u * Integral s(x)*dx = u**/3 + C1 is not known. Therefore, the numerical cell average is used: .. code-block:: none Snum(x) = Integral s(x)*dx = s(i) * (x[i]-x[i-1]). Numerical vs. manufactured solution plot (Nx=80): .. image:: _static/tutorial_cv_8-results.png :width: 500px The normalised global errors and the order of accuracy plots for the Koren flux limiter (grids 40, 60, 80, 120): .. image:: _static/tutorial_cv_8-results-koren.png :width: 800px """ import sys, math, numpy from time import localtime, strftime import matplotlib.pyplot as plt from daetools.pyDAE import * # Standard variable types are defined in variable_types.py from pyUnits import m, g, kg, s, K, mol, kmol, J, um c_t = daeVariableType("c_t", dimless, -1.0e+20, 1.0e+20, 0.0, 1e-07) L = 1.0 pi = numpy.pi tend = 0.5 t = Time() def c_exact(x,time): return numpy.sin(pi*(x-time))**2 / (1 - time*numpy.sin(pi*(x-time))**2) class modTutorial(daeModel): def __init__(self, Name, Parent = None, Description = ""): daeModel.__init__(self, Name, Parent, Description) self.x = daeDomain("x", self, m, "") self.c_exact = daeVariable("c_exact", c_t, self, "c using the analytical solution", [self.x]) self.c = daeVariable("c", c_t, self, "c using high resolution upwind scheme", [self.x]) def DeclareEquations(self): daeModel.DeclareEquations(self) xp = self.x.Points Nx = self.x.NumberOfPoints hr = daeHRUpwindSchemeEquation(self.c, self.x, daeHRUpwindSchemeEquation.Phi_Koren, 1e-10) c = lambda i: self.c(i) # Manufactured (exact) solution for i in range(0, Nx): eq = self.CreateEquation("c_exact(%d)" % i, "") eq.Residual = self.c_exact(i) - c_exact(xp[i],t) eq.CheckUnitsConsistency = False # The source function def s(i): return c(i)**2 # The analytical source term integral for consistent discretisation of convection and source terms: # S = 1/u * Integral(s(x)*dx) def S(i): C1 = 0.0 return c(i)**2 / 3 + C1 # Convection-diffusion equation for i in range(1, Nx): eq = self.CreateEquation("c(%d)" % i, "") eq.Residual = hr.dc_dt(i) + hr.dc_dx(i) - hr.source(s,i) eq.CheckUnitsConsistency = False # BCs eq = self.CreateEquation("c(0)", "") eq.Residual = c(0) - c_exact(xp[0], t) eq.CheckUnitsConsistency = False class simTutorial(daeSimulation): def __init__(self, Nx): daeSimulation.__init__(self) self.m = modTutorial("tutorial_cv_8(%d)" % Nx) self.m.Description = __doc__ self.Nx = Nx def SetUpParametersAndDomains(self): self.m.x.CreateStructuredGrid(self.Nx, 0.0, L) def SetUpVariables(self): Nx = self.m.x.NumberOfPoints xp = self.m.x.Points for i in range(1, Nx): self.m.c.SetInitialCondition(i, c_exact(xp[i], 0.0)) # Setup everything manually and run in a console def simulate(Nx): # Create Log, Solver, DataReporter and Simulation object log = daePythonStdOutLog() daesolver = daeIDAS() datareporter = daeDelegateDataReporter() simulation = simTutorial(Nx) # Do no print progress log.PrintProgress = False daesolver.RelativeTolerance = 1e-7 # Enable reporting of all variables simulation.m.SetReportingOn(True) # Set the time horizon and the reporting interval simulation.ReportingInterval = 0.05 simulation.TimeHorizon = tend # Connect data reporter simName = simulation.m.Name + strftime(" [%d.%m.%Y %H:%M:%S]", localtime()) # 1. TCP/IP tcpipDataReporter = daeTCPIPDataReporter() datareporter.AddDataReporter(tcpipDataReporter) if not tcpipDataReporter.Connect("", simName): sys.exit() # 2. Data dr = daeNoOpDataReporter() datareporter.AddDataReporter(dr) # Initialize the simulation simulation.Initialize(daesolver, datareporter, log) # Save the model report and the runtime model report #simulation.m.SaveModelReport(simulation.m.Name + ".xml") #simulation.m.SaveRuntimeModelReport(simulation.m.Name + "-rt.xml") # Solve at time=0 (initialization) simulation.SolveInitial() # Run simulation.Run() simulation.Finalize() ########################################### # Data # ########################################### results = dr.Process.dictVariables cvar = results[simulation.m.Name + '.c'] c_var_exact = results[simulation.m.Name + '.c_exact'] c = cvar.Values[-1, :] # 2D array [t,x] c_exact = c_var_exact.Values[-1, :] # 2D array [t,x] return c, c_exact def run(**kwargs): Nxs = numpy.array([40, 60, 80, 120]) n = len(Nxs) hs = L / Nxs E = numpy.zeros(n) C = numpy.zeros(n) p = numpy.zeros(n) E2 = numpy.zeros(n) # The normalised global errors for i,Nx in enumerate(Nxs): numerical_sol, manufactured_sol = simulate(int(Nx)) E[i] = numpy.sqrt((1.0/Nx) * numpy.sum((numerical_sol-manufactured_sol)**2)) # Order of accuracy for i,Nx in enumerate(Nxs): p[i] = numpy.log(E[i]/E[i-1]) / numpy.log(hs[i]/hs[i-1]) C[i] = E[i] / hs[i]**p[i] C2 = 17 # constant for the second order slope line (to get close to the actual line) E2 = C2 * hs**2 # E for the second order slope fontsize = 14 fontsize_legend = 11 fig = plt.figure(figsize=(10,4), facecolor='white') fig.canvas.set_window_title('The Normalised global errors and the Orders of accuracy (Nelems = %s) (cv_8)' % Nxs.tolist()) ax = plt.subplot(121) plt.figure(1, facecolor='white') plt.loglog(hs, E, 'ro', label='E(h)') plt.loglog(hs, E2, 'b-', label='2nd order slope') plt.xlabel('h', fontsize=fontsize) plt.ylabel('||E||', fontsize=fontsize) plt.legend(fontsize=fontsize_legend) #plt.xlim((0.04, 0.11)) ax = plt.subplot(122) plt.figure(1, facecolor='white') plt.semilogx(hs[1:], p[1:], 'rs-', label='Order of Accuracy (p)') plt.xlabel('h', fontsize=fontsize) plt.ylabel('p', fontsize=fontsize) plt.legend(fontsize=fontsize_legend) #plt.xlim((0.04, 0.075)) #plt.ylim((2.0, 2.04)) plt.tight_layout() plt.show() if __name__ == "__main__": run()