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#!/usr/bin/env python
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"""
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A simple WaveSolver class for evolving the wave equation in 2D.
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This works in parallel by using a RectPartitioner object.
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Authors
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-------
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* Xing Cai
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* Min Ragan-Kelley
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"""
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import time
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from numpy import exp, zeros, newaxis, sqrt, arange
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def iseq(start=0, stop=None, inc=1):
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"""
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Generate integers from start to (and including!) stop,
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with increment of inc. Alternative to range/xrange.
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"""
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if stop is None: # allow isequence(3) to be 0, 1, 2, 3
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# take 1st arg as stop, start as 0, and inc=1
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stop = start; start = 0; inc = 1
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return arange(start, stop+inc, inc)
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class WaveSolver(object):
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"""
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Solve the 2D wave equation u_tt = u_xx + u_yy + f(x,y,t) with
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u = bc(x,y,t) on the boundary and initial condition du/dt = 0.
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Parallelization by using a RectPartitioner object 'partitioner'
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nx and ny are the total number of global grid cells in the x and y
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directions. The global grid points are numbered as (0,0), (1,0), (2,0),
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..., (nx,0), (0,1), (1,1), ..., (nx, ny).
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dt is the time step. If dt<=0, an optimal time step is used.
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tstop is the stop time for the simulation.
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I, f are functions: I(x,y), f(x,y,t)
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user_action: function of (u, x, y, t) called at each time
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level (x and y are one-dimensional coordinate vectors).
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This function allows the calling code to plot the solution,
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compute errors, etc.
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implementation: a dictionary specifying how the initial
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condition ('ic'), the scheme over inner points ('inner'),
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and the boundary conditions ('bc') are to be implemented.
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Normally, values are legal: 'scalar' or 'vectorized'.
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'scalar' means straight loops over grid points, while
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'vectorized' means special NumPy vectorized operations.
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If a key in the implementation dictionary is missing, it
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defaults in this function to 'scalar' (the safest strategy).
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Note that if 'vectorized' is specified, the functions I, f,
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and bc must work in vectorized mode. It is always recommended
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to first run the 'scalar' mode and then compare 'vectorized'
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results with the 'scalar' results to check that I, f, and bc
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work.
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verbose: true if a message at each time step is written,
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false implies no output during the simulation.
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final_test: true means the discrete L2-norm of the final solution is
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to be computed.
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"""
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def __init__(self, I, f, c, bc, Lx, Ly, partitioner=None, dt=-1,
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user_action=None,
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implementation={'ic': 'vectorized', # or 'scalar'
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'inner': 'vectorized',
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'bc': 'vectorized'}):
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nx = partitioner.global_num_cells[0] # number of global cells in x dir
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ny = partitioner.global_num_cells[1] # number of global cells in y dir
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dx = Lx/float(nx)
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dy = Ly/float(ny)
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loc_nx, loc_ny = partitioner.get_num_loc_cells()
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nx = loc_nx; ny = loc_ny # now use loc_nx and loc_ny instead
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lo_ix0 = partitioner.subd_lo_ix[0]
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lo_ix1 = partitioner.subd_lo_ix[1]
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hi_ix0 = partitioner.subd_hi_ix[0]
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hi_ix1 = partitioner.subd_hi_ix[1]
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x = iseq(dx*lo_ix0, dx*hi_ix0, dx) # local grid points in x dir
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y = iseq(dy*lo_ix1, dy*hi_ix1, dy) # local grid points in y dir
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self.x = x
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self.y = y
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xv = x[:,newaxis] # for vectorized expressions with f(xv,yv)
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yv = y[newaxis,:] # -- " --
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if dt <= 0:
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dt = (1/float(c))*(1/sqrt(1/dx**2 + 1/dy**2)) # max time step
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Cx2 = (c*dt/dx)**2; Cy2 = (c*dt/dy)**2; dt2 = dt**2 # help variables
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u = zeros((nx+1,ny+1)) # solution array
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u_1 = u.copy() # solution at t-dt
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u_2 = u.copy() # solution at t-2*dt
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# preserve for self.solve
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implementation=dict(implementation) # copy
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if 'ic' not in implementation:
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implementation['ic'] = 'scalar'
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if 'bc' not in implementation:
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implementation['bc'] = 'scalar'
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if 'inner' not in implementation:
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implementation['inner'] = 'scalar'
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self.implementation = implementation
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self.Lx = Lx
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self.Ly = Ly
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self.I=I
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self.f=f
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self.c=c
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self.bc=bc
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self.user_action = user_action
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self.partitioner=partitioner
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# set initial condition (pointwise - allows straight if-tests in I(x,y)):
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t=0.0
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if implementation['ic'] == 'scalar':
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for i in xrange(0,nx+1):
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for j in xrange(0,ny+1):
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u_1[i,j] = I(x[i], y[j])
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for i in xrange(1,nx):
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for j in xrange(1,ny):
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u_2[i,j] = u_1[i,j] + \
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0.5*Cx2*(u_1[i-1,j] - 2*u_1[i,j] + u_1[i+1,j]) + \
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0.5*Cy2*(u_1[i,j-1] - 2*u_1[i,j] + u_1[i,j+1]) + \
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dt2*f(x[i], y[j], 0.0)
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# boundary values of u_2 (equals u(t=dt) due to du/dt=0)
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i = 0
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for j in xrange(0,ny+1):
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u_2[i,j] = bc(x[i], y[j], t+dt)
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j = 0
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for i in xrange(0,nx+1):
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u_2[i,j] = bc(x[i], y[j], t+dt)
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i = nx
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for j in xrange(0,ny+1):
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u_2[i,j] = bc(x[i], y[j], t+dt)
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j = ny
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for i in xrange(0,nx+1):
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u_2[i,j] = bc(x[i], y[j], t+dt)
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elif implementation['ic'] == 'vectorized':
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u_1 = I(xv,yv)
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u_2[1:nx,1:ny] = u_1[1:nx,1:ny] + \
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0.5*Cx2*(u_1[0:nx-1,1:ny] - 2*u_1[1:nx,1:ny] + u_1[2:nx+1,1:ny]) + \
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0.5*Cy2*(u_1[1:nx,0:ny-1] - 2*u_1[1:nx,1:ny] + u_1[1:nx,2:ny+1]) + \
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dt2*(f(xv[1:nx,1:ny], yv[1:nx,1:ny], 0.0))
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# boundary values (t=dt):
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i = 0; u_2[i,:] = bc(x[i], y, t+dt)
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j = 0; u_2[:,j] = bc(x, y[j], t+dt)
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i = nx; u_2[i,:] = bc(x[i], y, t+dt)
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j = ny; u_2[:,j] = bc(x, y[j], t+dt)
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if user_action is not None:
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user_action(u_1, x, y, t) # allow user to plot etc.
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# print list(self.us[2][2])
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self.us = (u,u_1,u_2)
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def solve(self, tstop, dt=-1, user_action=None, verbose=False, final_test=False):
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t0=time.time()
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f=self.f
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c=self.c
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bc=self.bc
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partitioner = self.partitioner
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implementation = self.implementation
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nx = partitioner.global_num_cells[0] # number of global cells in x dir
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ny = partitioner.global_num_cells[1] # number of global cells in y dir
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dx = self.Lx/float(nx)
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dy = self.Ly/float(ny)
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loc_nx, loc_ny = partitioner.get_num_loc_cells()
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nx = loc_nx; ny = loc_ny # now use loc_nx and loc_ny instead
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x = self.x
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y = self.y
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xv = x[:,newaxis] # for vectorized expressions with f(xv,yv)
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yv = y[newaxis,:] # -- " --
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if dt <= 0:
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dt = (1/float(c))*(1/sqrt(1/dx**2 + 1/dy**2)) # max time step
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Cx2 = (c*dt/dx)**2; Cy2 = (c*dt/dy)**2; dt2 = dt**2 # help variables
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# id for the four possible neighbor subdomains
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lower_x_neigh = partitioner.lower_neighbors[0]
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upper_x_neigh = partitioner.upper_neighbors[0]
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lower_y_neigh = partitioner.lower_neighbors[1]
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upper_y_neigh = partitioner.upper_neighbors[1]
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u,u_1,u_2 = self.us
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# u_1 = self.u_1
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t = 0.0
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while t <= tstop:
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t_old = t; t += dt
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if verbose:
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print 'solving (%s version) at t=%g' % \
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(implementation['inner'], t)
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# update all inner points:
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if implementation['inner'] == 'scalar':
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for i in xrange(1, nx):
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for j in xrange(1, ny):
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u[i,j] = - u_2[i,j] + 2*u_1[i,j] + \
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Cx2*(u_1[i-1,j] - 2*u_1[i,j] + u_1[i+1,j]) + \
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Cy2*(u_1[i,j-1] - 2*u_1[i,j] + u_1[i,j+1]) + \
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dt2*f(x[i], y[j], t_old)
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elif implementation['inner'] == 'vectorized':
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u[1:nx,1:ny] = - u_2[1:nx,1:ny] + 2*u_1[1:nx,1:ny] + \
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Cx2*(u_1[0:nx-1,1:ny] - 2*u_1[1:nx,1:ny] + u_1[2:nx+1,1:ny]) + \
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Cy2*(u_1[1:nx,0:ny-1] - 2*u_1[1:nx,1:ny] + u_1[1:nx,2:ny+1]) + \
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dt2*f(xv[1:nx,1:ny], yv[1:nx,1:ny], t_old)
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# insert boundary conditions (if there's no neighbor):
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if lower_x_neigh < 0:
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if implementation['bc'] == 'scalar':
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i = 0
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for j in xrange(0, ny+1):
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u[i,j] = bc(x[i], y[j], t)
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elif implementation['bc'] == 'vectorized':
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u[0,:] = bc(x[0], y, t)
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if upper_x_neigh < 0:
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if implementation['bc'] == 'scalar':
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i = nx
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for j in xrange(0, ny+1):
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u[i,j] = bc(x[i], y[j], t)
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elif implementation['bc'] == 'vectorized':
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u[nx,:] = bc(x[nx], y, t)
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if lower_y_neigh < 0:
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if implementation['bc'] == 'scalar':
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j = 0
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for i in xrange(0, nx+1):
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u[i,j] = bc(x[i], y[j], t)
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elif implementation['bc'] == 'vectorized':
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u[:,0] = bc(x, y[0], t)
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if upper_y_neigh < 0:
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if implementation['bc'] == 'scalar':
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j = ny
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for i in xrange(0, nx+1):
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u[i,j] = bc(x[i], y[j], t)
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elif implementation['bc'] == 'vectorized':
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u[:,ny] = bc(x, y[ny], t)
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# communication
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partitioner.update_internal_boundary (u)
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if user_action is not None:
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user_action(u, x, y, t)
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# update data structures for next step
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u_2, u_1, u = u_1, u, u_2
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t1 = time.time()
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print 'my_id=%2d, dt=%g, %s version, slice_copy=%s, net Wtime=%g'\
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%(partitioner.my_id,dt,implementation['inner'],\
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partitioner.slice_copy,t1-t0)
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# save the us
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self.us = u,u_1,u_2
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# check final results; compute discrete L2-norm of the solution
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if final_test:
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loc_res = 0.0
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for i in iseq(start=1, stop=nx-1):
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for j in iseq(start=1, stop=ny-1):
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loc_res += u_1[i,j]**2
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return loc_res
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return dt
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