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Diagonalization of 1-D differential operators with piecewise constant coefficients using the uncertainty principle

Author

Listed:
  • Long, Sarah D.
  • Sheikholeslami, Somayyeh
  • Lambers, James V.
  • Walker, Carley

Abstract

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 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.

Suggested Citation

  • Long, Sarah D. & Sheikholeslami, Somayyeh & Lambers, James V. & Walker, Carley, 2019. "Diagonalization of 1-D differential operators with piecewise constant coefficients using the uncertainty principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 194-226.
    • Handle: RePEc:eee:matcom:v:156:y:2019:i:c:p:194-226
    DOI: 10.1016/j.matcom.2018.08.003
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