A highly accurate and efficient numerical method is presented for computing the solution of a 1-D time-dependent partial differential equation in which the spatial differential operator features a piecewise constant coefficient defined on n" role="presentation"> pieces, in either self-adjoint and non-self-adjoint form, on a finite interval with periodic boundary conditions. The Uncertainty Principle is used to estimate the eigenvalues of the operator. Then, these estimates are used to construct a basis of eigenfunctions for use with a spectral method. The solution is presented as a truncated eigenfunction expansion, where each eigenfunction is a wave function that changes frequencies at the interfaces between different materials. Numerical experiments demonstrate the accuracy, efficiency and scalability of the method in comparison to other methods.
Mathematics and Computers in Simulation
Long, S. D.,
Lambers, J. V.,
(2019). Diagonalization of 1-D Differential Operators With Piecewise Constant Coefficients Using the Uncertainty Principle. Mathematics and Computers in Simulation, 156, 194-226.
Available at: https://aquila.usm.edu/fac_pubs/15542
Available for download on Tuesday, September 01, 2020