[SciPy-dev] Test case for integrate/ode.py: banded systems

Andrew Straw strawman at astraw.com
Mon Mar 6 10:57:09 CST 2006


Hi Jesper,

Thanks, this looks interesting. I've copied your email to the following
location so it doesn't slip through the cracks:

http://www.scipy.org/Developer_Zone/Recipes/ODE_Integration_on_Banded_Systems

You indicated it might serve as an example, so I hope after any
potential refinement in the "Developer Zone" you (or someone with more
knowledge of integration on banded systems than I) can move it over to
the "Cookbook".

Cheers!
Andrew

Jesper Friis wrote:

> I hope this is the correct place to post messages like this. Last
> Friday I submitted a small patch for integrate/ode.py making it
> working for banded systems. I have now successfully applied the solver
> to the banded example problem included in the cvode package. The
> attached script might both work as a test and as an example on how to
> solve banded systems.
>
> I think, especially the following comment considering the Jacobian
> might be of general interest:
>     # The Jacobian.
>     # For banded systems this function returns a matrix pd of
>     # size ml+mu*2+1 by neq containing the partial derivatives
>     # df[k]/du[u]. Here f is the right hand side function in
>     # udot=f(t,u), ml and mu are the lower and upper half bandwidths
>     # and neq the number of equations.  The derivatives df[k]/du[l]
>     # are loaded into pd[mu+k-l,k], i.e. the diagonals are loaded into
>     # the rows of pd from top down (fortran indexing).
>     #
>     # Confusingly, the number of rows VODE expect is not ml+mu+1, but
>     # given by a parameter nrowpd, which unfortunately is left out in
>     # the python interface. However, it seems that VODE expect that
>     # nrowpd = ml+2*mu+1. E.g. for our system with ml=mu=5 VODE expect
>     # 16 rows. Fortunately the f2py interface prints out an error if
>     # the number of rows is wrong, so as long ml and mu are known in
>     # beforehand one can always determine nrowpd by trial and error...
>
>
>
> Regards
> /Jesper
>
>------------------------------------------------------------------------
>
>#!/usr/bin/env python
>
># Test provided provided by Jesper Friis, <jesper.friis at material.ntnu.no>
>
>from scipy import arange,zeros,exp
>from scipy.integrate import ode
>
>def test_ode_banded():
>    """Banded example copied from CVODE.
>    
>    The following is a simple example problem with a banded Jacobian,
>    with the program for its solution by CVODE.
>    The problem is the semi-discrete form of the advection-diffusion 
>    equation in 2-D:                                                 
>      du/dt = d^2 u / dx^2 + 0.5 du/dx + d^2 u / dy^2                 
>    on the rectangle 0 <= x <= 2, 0 <= y <= 1, and the time          
>    interval 0 <= t <= 1.  Homogeneous Dirichlet boundary conditions 
>    are posed, and the initial condition is                          
>      u(x,y,t=0) = x(2-x)y(1-y)exp(5xy) .                            
>    The PDE is discretized on a uniform MX+2 by MY+2 grid with       
>    central differencing, and with boundary values eliminated,       
>    leaving an ODE system of size NEQ = MX*MY.                       
>
>    Assuming that MY < MX a minimum bandwidth banded system can be
>    constructed by arranging the grid points along columns. This
>    results in the lower and upper bandwidths ml = mu = MY.
>
>    This function solves the problem with the BDF method, Newton      
>    iteration with the VODE band linear solver, and a user-supplied 
>    Jacobian routine. It uses scalar relative and absolute tolerances.
>    Output is printed at t = 0., .1, .2, ..., 1.                         
>    """
>    # Some constants
>    xmax = 2.0              # domain boundaries
>    ymax = 1.0
>    mx = 10                 # mesh dimensions
>    my = 5
>    dx = xmax/(mx+1.)       # grid spacing
>    dy = ymax/(my+1.)
>    neq = mx*my             # number of equations
>    mu = my                 # half bandwidths
>    ml = my
>    atol = 1.e-5            # scalar absolute tolerance
>    nrowpd = ml+2*mu+1      # number of rows in storage of banded Jacobian
>    x = dx*(arange(mx)+1.0) # inner grid points
>    y = dy*(arange(my)+1.0)
>    t = 0.1*arange(11)      # the times we want to print the solution
>
>    # The right hand side function in udot = f(t,u)
>    def f(t,u):
>        for j in range(mx):
>            for i in range(my):
>                # Get index of gridpoint i,j in u and udot
>                k = j*my+i 
>                # Extract u at x_j, y_i and four neighboring points
>                ult = urt = uup = udn = 0.0
>                uij = u[k]
>                if j>0:    ult = u[k-my]
>                if j<mx-1: urt = u[k+my]
>                if i>0:    uup = u[k-1]
>                if i<my-1: udn = u[k+1]
>                # Set diffusion and advection terms and load into udot
>                hdiff = (ult - 2.0*uij + urt)/(dx*dx)
>                vdiff = (uup - 2.0*uij + udn)/(dy*dy)
>                hadv  = 0.5*(urt - ult)/(2.0*dx)
>                udot[j*my+i] = hdiff + hadv + vdiff
>        return udot
>
>
>    # The Jacobian.
>    # For banded systems this function returns a matrix pd of
>    # size ml+mu*2+1 by neq containing the partial derivatives
>    # df[k]/du[u]. Here f is the right hand side function in
>    # udot=f(t,u), ml and mu are the lower and upper half bandwidths
>    # and neq the number of equations.  The derivatives df[k]/du[l]
>    # are loaded into pd[mu+k-l,k], i.e. the diagonals are loaded into
>    # the rows of pd from top down (fortran indexing).
>    #
>    # Confusingly, the number of rows VODE expect is not ml+mu+1, but
>    # given by a parameter nrowpd, which unfortunately is left out in
>    # the python interface. However, it seems that VODE expect that
>    # nrowpd = ml+2*mu+1. E.g. for our system with ml=mu=5 VODE expect
>    # 16 rows. Fortunately the f2py interface prints out an error if
>    # the number of rows is wrong, so as long ml and mu are known in
>    # beforehand one can always determine nrowpd by trial and error...
>    def jac(t,u):
>        # The components of u that f[i,j] = udot_ij depends on are:
>        # u[i,j], u[i,j-1], u[i,j+1], u[i-1,j] and u[i+1,j], with
>        #   df[i,j]/du[i,j]   = -2 (1/dx^2 + 1/dy^2),          l=k
>        #   df[i,j]/du[i,j-1] = 1/dx^2 - .25/dx,       j>0,    l=k-my
>        #   df[i,j]/du[i,j+1] = 1/dx^2 + .25/dx,       j<MX-1, l=k+my
>        #   df[i,j]/du[i-1,j] = 1/dy^2                 i>0,    l=k-1
>        #   df[i,j]/du[i+1,j] = 1/dy^2                 i<MY-1, l=k+1
>        # where k=j*my+i.
>        for j in range(mx):
>            for i in range(my):
>                k = j*my+i
>                pd[mu,k] = -2.0*(1.0/(dx*dx) + 1.0/(dy*dy))
>                if j > 0:    pd[mu-my,k] = 1.0/(dx*dx) + 0.25/dx
>                if j < mx-1: pd[mu+my,k] = 1.0/(dx*dx) + 0.25/dx
>                if i > 0:    pd[mu-1,k]  = 1.0/(dy*dy)
>                if k < my-1: pd[mu+1,k]  = 1.0/(dy*dy)
>        return pd
>
>    # Initial value
>    u = zeros(neq,float)
>    for j in range(mx):
>        u[j*my:(j+1)*my] = x[j]*(xmax - x[j])*y*(ymax - y)*exp(5*x[j]*y)
>
>    # Allocate global work arrays pd and udot
>    pd   = zeros((nrowpd,neq),float)
>    udot = zeros(neq,float)
>
>    # Solve the problem
>    print "2-D advection-diffusion equation, mesh dimensions =%3d %3d" %(mx,my)
>    print "Banded solution, bandwidth = %d" % (ml+mu+1)
>    r = ode(f, jac)
>    r.set_integrator('vode',atol=atol,lband=ml,uband=mu,method='bdf')
>    r.set_initial_value(u, t=t[0])
>    print 'At t=%4.2f    max.norm(u) = %-12.4e'%(r.t, max(u))
>    for tout in t[1:]:
>        u = r.integrate(tout)
>        print 'At t=%4.2f    max.norm(u) = %-12.4e'%(r.t, max(u))
>        if not r.successful():
>            print "An error occurred during integration"
>            break
>    
>
>
>test_ode_banded()
>  
>
>------------------------------------------------------------------------
>
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