1. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Overlapping Multidomain Chebyshev Method: Verification

2. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Seismic wave Propagation and Imaging in Complex media: a European network PETER DANECEK Early Stage Researcher Host Institution: INGV Bologna, Italy Place of Origin: Zlin, Czech Republic Appointment Time: Juni 2006 Project: Multidomain Chebyshev Method for Wave Propagation on Continental and Global Scales with Application. Task Groups: TG Numerical Methods, TG Global Scale Cooperation: OGS Trieste, Munich University PETER DANECEK Early Stage Researcher Host Institution: INGV Bologna, Italy Place of Origin: Zlin, Czech Republic Appointment Time: Juni 2006 Project: Multidomain Chebyshev Method for Wave Propagation on Continental and Global Scales with Application. Task Groups: TG Numerical Methods, TG Global Scale Cooperation: OGS Trieste, Munich University Profile

3. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Overview Motivation and design goals Numerical method: OMDC Geometry Results Future work

4. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Design goals Parallel 3D method with good scalability Efficient, structured method => implicit topology Continental & Global Scale Seismology => spherical geometry Justification: - current Earth models are smooth - grid generation AND inversion

5. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Numerical method (1) N = 6 Overlapping Multidomain Chebyshev Method

6. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Numerical method (2) N = 6 OMDC

7. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Numerical method How does it relate to other methods? OMDC ? ? ? ? ? ?

8. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Numerical method How does it relate to other methods? OMDC low order FD higher order FD SEM FEM with high order ordinary polynomials low order FEM Increasing order Strong formulation Weak formulation global local Fourier method equal-spaced Chebyshev method special co-location

9. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Spherical geometry (1) Uniform mapping by continuous functions for the whole computational domain Elastic equations solve directly in spherical coordinate system Orthogonality between coordinate lines Few pre-calculated/stored parameter

10. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Spherical geometry (2) Sphere decomposed into six equally shaped chunks Each chunk has a uniform mapping by continuous functions Coordinates lines along the surface are no longer orthogonal Derivatives are coupled Elastic equations solve directly in the resulting coordinate systems Still moderate number of pre- calculated/stored parameter

11. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Results (1) 8 processors N = 8 32x32x40 elements 226x226x282 points Unity sphere with r= 0.8..1.0,  =  = -5..+5 degree Grid spacing: 0.17..1.08 *10 -3 Vs = 0.1, Vp = 0.2 ca. 10 ppw, c < 0.1, leap-frog Source: point force in radial direction Absorbing boundaries

12. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Results (1)

13. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Results (2) 64 processors N = 8 32x32x40 elements 226x226x282 points Unity sphere with r= 0.8..1.0,  =  = -5..+5 degree Grid spacing: 0.17..1.08 *10 -3 Vs = 0.1, Vp = 0.2 ca. 10 ppw, c < 0.1, leap-frog Source: Explosion (stress) Absorbing boundaries

14. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Results (2)

15. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Results model 1 model 2

16. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Future work Time histories Comparison against known solution Boundary conditions Complex models Code extensions and improvements Test the limits Comparison with other methods Applications More work on Cubed Sphere geometry