JAMP  Vol.3 No.3 , March 2015
Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System
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.
Cite this paper: Zhou, B. , Heinson, G. and Rivera-Rios, A. (2015) Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System. Journal of Applied Mathematics and Physics, 3, 358-370. doi: 10.4236/jamp.2015.33047.

[1]   Pysklywec, R.N. and Cruden, A.R. (2004) Coupled Crust-Mantle Dynamics and Inter-Plate Tectonics: Two-Dimensional Numerical and Three-Dimensional Analogue Modeling. Geochemistry Geophysics Geosystems, 5, 1-20.

[2]   Taras, G. (2010) Introduction to Numerical Geodynamic Modeling. Cambridge University Press, Cambridge.

[3]   Harbaugh, A.W. and McDonald, M.G. (1996) User’s Documentation for MODFLOW-96, an Updata to the US Geological Survey Modular Finite-Difference Ground-Water Flow. US Geological Survey Open-File Report 96-485.

[4]   Dierch, H.-J.G. (1998) FEFLOW-User’s Manual: WASI-Institute of Water Resources. Planning and System Research Ltd., Berlin.

[5]   Bundschuh, J. and Arriage, M.C.S. (2010) Introduction to the Numerical Modelling of Groundwater and Geothermal Systems, Fundamentals of Mass, Energy and Solute Transport in Poroelastic Rocks. CRC Press, Taylor & Francis Group, Boca Raton.

[6]   Kelley, K.R. and Marfurt, K.J. (1990) Numerical Solutions of Acoustic and Elastic Wave Equations: Finite-Difference and Finite-Element Algorithms. Society of Exploration Geophysics, Tulsa.

[7]   Kerry, K. and Weiss, C. (2006) Adaptive Finite Element Modeling Using Unstructuredgrid, the 2D Magnetotelluric Example. Geophysics, 71, G291-G294.

[8]   Zhou, B. and Greenhalgh, S. (2011) 3-D Frequency-Domain Seismic Wave Modeling in Heterogeneous, Anisotropic Media Using a Gaussian Quadrature Grid Approach. Geophysical Journal International, 184, 507-526.

[9]   Zhou, B., Greenhalgh, S. and Maurer, H. (2012) 2.5-D Frequency-Domain Seismic Wave Modeling in Heterogeneous, Anisotropic Media Using a Gaussian Quadrature Grid Technique. Computer & Geosciences, 39, 18-33.

[10]   Robert, T., Vivier, F. and Shenghui, L. (2004) Three-Dimensional Modelling of Ocean Electrodynamics Using Gauged Potentials. Geophysical Journal International, 158, 874-887.

[11]   Virieux, J., Calandra, H. and Plessix, R. (2011) A Review of the Spectral, Pseudo-Spectral, Finite-Difference and Finite-Element Modeling Techniques for Geophysical Imaging. Geophysical Prospecting, 59, 794-813.

[12]   Dablain, M.A. (1986) The Application of High-Order Differencing to the Scalar Wave Equation. Geophysics, 51, 54-66.

[13]   Gilles, L., Hagness, S.C. and Vazquez, L. (2000) Comparison between Staggered and Unstaggered Finite-Difference Time-Domain Grid for Few-Cycle Temporal Optical Solution Propagation. Journal of Computational Physics, 161, 379-400.

[14]   Festa, G. and Vilotte, J. (2005) The Newmark Scheme as Velocity-Stress Time Staggered: An Efficient PML Implementation for Spectral Element Simulations of Electrodynamics. Geophysical Journal International, 161, 789-812.

[15]   Streich, R. (2009) 3D Finite-Difference Frequency-Domain Modeling of Controlled-Source Electromagnetic Data: Direct Solution and Optimization for High Accuracy. Geophysics, 74, F95-F105.

[16]   Robertsson, J.O. (1996) Numerical Free-Surface Condition for Elastic/Viscoelastic Finite-Difference Modeling in the Presence of Topography. Geophysics, 61, 1921-1934.

[17]   Schwarz, H.R. (1988) Finite Element Methods. Academic Press, New York.

[18]   Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Methods in Fluid Dynamics. Springer-Verlag, New York.

[19]   Komatitsch, D. and Tromp, J. (2002) Spectral-Element Simulation of Global Seismic Wave Propagation—I. Validation. Geophysical Journal International, 149, 390-412.

[20]   Komatitsch, D., Coutel, F. and Mora, P. (1996) Tensorial Formulation of the Wave-Equation for Modeling Curved Interfaces. Geophysical Journal International, 127, 156-168.

[21]   Hestholm, S. and Ruud, B. (1998) 3-D Finite-Difference Elastic Wave Modeling Including Surface Topography. Geophysics, 63, 613-622.

[22]   Zhang, W. and Chen, X. (2006) Traction Image Method for Irregular Free Surface Boundaries in Finite-Difference Seismic Wave Simulation. Geophysical Journal International, 167, 337-353.

[23]   Trefethen, L.N. (2000) Spectral Method in MATLAB. SIAM, Philadelphia.

[24]   Tessmer, E. and Kosloff, D. (1994) 3-D Elastic Modeling with Surface Topography by a Chebyshev Spectral Method. Geophysics, 59, 464-473.

[25]   Igel, H. (1999) Wave Propagation in a Three-Dimensional Spherical Section by the Chebyshev Spectral Method. Geophysical Journal International, 136, 559-566.

[26]   John, F. (1982) Partial Differential Equations. Springer-Verlag, New York.

[27]   Vorst, H.A. (2003) Iterative Krylov Methods for a Large Linear Systems. Cambridge University Press, Cambridge.

[28]   Amestoy, P.R., Guermouche, A., L’Excellent, J.-Y. and Pralet, S. (2006) Hybrid Scheduling for the Parallel Solution of Linear Systems. Parallel Computing, 32, 136-156.

[29]   Helmuth, S. (1995) One Dimensional Spline Interpolation Algorithms. A. K. Peters, Wellesley.

[30]   Helmuth, S. (1995) Two Dimensional Spline Interpolation Algorithms. A. K. Peters, Wellesley.