TITLE:
Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System
AUTHORS:
Bing Zhou, Graham Heinson, Aixa Rivera-Rios
KEYWORDS:
Numerical Differentiation, Chebyshev Spectral Method, Curved Coordinate System, Arbitrary Topography
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.3 No.3,
March
30,
2015
ABSTRACT: A new
numerical approach, called the “subdomain Chebyshev spectral method” is
presented for calculation of the spatial derivatives in a curved coordinate
system, which may be employed for numerical solutions of partial differential
equations defined in a 2D or 3D geological model. The new approach refers to a “strong
version” against the “weak version” of the subspace spectral method based on
the variational principle or Galerkin’s weighting scheme. We incorporate local
nonlinear transformations and global spline interpolations in a curved
coordinate system and make the discrete grid exactly matches geometry of the
model so that it is achieved to convert the global domain into subdomains and
apply Chebyshev points to locally sampling physical quantities and globally
computing the spatial derivatives. This new approach not only remains
exponential convergence of the standard spectral method in subdomains, but also
yields a sparse assembled matrix when applied for the global domain
simulations. We conducted 2D and 3D synthetic experiments and compared
accuracies of the numerical differentiations with traditional finite difference
approaches. The results show that as the points of differentiation vector are
larger than five, the subdomain Chebyshev spectral method significantly improve
the accuracies of the finite difference approaches.