Lokálne chyby numerických schém

Transcription

1 Slovenská geofyzikálna konferencia 211 Lokálne chyby numerických schém Peter Moczo Jozef Kristek Martin Galis Emmanuel Chaljub, Vincent Etienne FMFI UK Bratislava GFÚ SAV Bratislava ISTerre Université Joseph Fourier Grenoble, France

2 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

3 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

4 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

5 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

6 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

7 3D numerical schemes method equation formulation grid add. specif. order FD D CG 2 = FE L8 FD DS PSG 2 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

8 method 3D numerical schemes equation formulation grid add. specif. order FD D CG 2 = FE L8 FD DS PSG 2 = FE G1 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

9 method 3D numerical schemes equation formulation grid add. specif. order FD D CG 2 = FE L8 FD DS PSG 2 = FE G1 partly FE L8 Lobatto 2 FE G1 finiteelement 1-point integr. FE G8 = DG CF 2 DG CF 2 discontinuous Galerkin centered flux FD D CG 4a 4 SE 4 cn, vn spectralelement GLL integr.

10 an unbounded homogeneous isotropic elastic medium and a uniform cubic grid all schemes in a unified form: { } (,, ; +Δ ) = numerical_scheme ( Δ ), ( ) U x y z t t U t t U t

11 numerical solution in one time step { } (,, ; +Δ ) = numerical_scheme ( Δ ), ( ) N E E U x y z t t U t t U t exact values for plane S wave

12 a relative local error in amplitude in onetimestep N A = numerical amplitude at t+ Δt E A = exact amplitude at t+ Δt ε Rel ampl 2 N E Δt ref A A = E Δt A Δt ref = Δt for p=.9 s= 1 6 V V = 1.42 P S

13 a relative local error in the vector difference in onetimestep N i U = numerical component at t+ Δt E U i = exact component at t+ Δt ε Rel Δtref 1 N E N E N E vdiff = E ( Ux Ux ) + ( Uy Uy ) + ( Uz Uz ) Δt A Δt ref = Δt for p=.9 s= 1 6 V V = 1.42 P S

14 solid circle corresponds to zero error x z

15 FD DS PSG 2 = FE G FE G8 = DG CF relative error in amplitude x FD D CG 4a SE 4 cn 18 SE 4 vn FD D CG 2 = FE L z spatial sampling : 2 nd -order schemes: 12 4 th -order schemes: 6 = 1.42 = 5 = 1

16 FD DS PSG 2 = FE G FE G8 = DG CF relative error in amplitude body diag FD D CG 4a SE 4 cn 18 SE 4 vn FD D CG 2 = FE L z spatial sampling : 2 nd -order schemes: 12 4 th -order schemes: 6 = 1.42 = 5 = 1

17 FD DS PSG 2 = FE G FE G8 = DG CF relative error in vector difference x FD D CG 4a SE 4 cn 18 SE 4 vn FD D CG 2 = FE L z spatial sampling : 2 nd -order schemes: 12 4 th -order schemes: 6 = 1.42 = 5 = 1

18 FD DS PSG 2 = FE G FE G8 = DG CF relative error in vector difference body diag FD D CG 4a SE 4 cn 18 SE 4 vn FD D CG 2 = FE L z spatial sampling : 2 nd -order schemes: 12 4 th -order schemes: 6 = 1.42 = 5 = 1

19 equivalent sampling equivalent sampling 1 FD D CG 2 = FE L p = p =.9 FE G8 = DG CF 2 FD DS PSG 2 = FEG1 SE 4 vn SE 4 cn equivalent spatial sampling based on FD D CG 4a relative error in amplitude FD DS PSG 2 = FEG1 FE G8 = DG CF 2 SE 4 vn SE 4 cn FD D CG 4a equivalent spatial sampling based on relative error in vector difference

20 results and conclusions the numerical amplitude is almost independent on the /V S ratio, if the discrete approximations to the 2 nd mixed and non-mixed spatial derivatives have the same coefficients of the leading terms of truncation errors this is the case of, FD D CG 4a, and FD DS PSG 2 = FE G1 otherwise the error in amplitude increases with increasing /V S ratio

21 results and conclusions for all schemes the error in the vector difference between the numerical and exact s increases with increasing /V S ratio this has to be accounted for by a proper spatial sampling

22 results and conclusions FD D CG 2 = FE L8 is the most sensitive to the increasing /V S ratio and for /V S > 2 requires considerably denser spatial sampling than any other scheme in order to achieve the same accuracy the maximum error of, FD DS PSG 2 = FE G1 and FE G8 = DG CF 2 increases in the same way FD DS PSG 2 = FE G1 and FE G8 = DG CF 2 require denser spatial sampling than in order to achieve the same accuracy

23 results and conclusions the maximum error of the 4th-order schemes increases in the same way FD D CG 4a,, SE 4 cn and SE 4 vn require denser spatial sampling than in order to achieve the same accuracy the 4 th -order schemes are for /V S > 3 less sensitive to the increasing /V S ratio than the 2 nd -order schemes