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To solve a scalar wave equation using space-time discretization of the action.
All the results are computed on a 4x4 Chebyshev grid.
To test the implementation of the integration matrix, and then the boundary projection operator by solving the Laplace equation as a test case.
Used two different methods to implement BCs:
u[0] = u[-1] = 0
u[:, 0] = u[:, -1] = 1
Here's the solution which looks very wrong:
Tried to implement BCs using a mixed approach:
- Dirichlet BCs on three sides using Lagrange multipliers
- Neumann BCs using the projection operator.
FIXME: However, multiplying the matrix with the projection operator leads to a singular matrix!
Hence, didn't multiply with the projection operator, but kept the 'b' vector intact. Here's the solution, which looks deceptively correct.
Grid size '4 x 4'
u(0) = np.exp(-(x)**2.0/2.0*25.0)
\dot(u)(0) = -0.004*x*np.exp(-(x)**2.0/2.0*25.0)
For understanding the motion, look at the plots here.
