1. Grid Coupling in TIMCOM 鄭偉明 TAY Wee-Beng Department of Atmospheric Sciences National Taiwan University

2. Motivation Multi-domain problems Multi-domain problems Problems which require 2 or more domains (grids) to solve efficiently. Problems which require 2 or more domains (grids) to solve efficiently. Arises due to difference in topography or restriction in computation resources. Arises due to difference in topography or restriction in computation resources.

3. Objective To transfer values from grid A to B and vice versa efficiently and conservatively To transfer values from grid A to B and vice versa efficiently and conservatively Efficiently Efficiently Algorithm must not take up too much computational resources. Algorithm must not take up too much computational resources. Relatively simple to program. Relatively simple to program. Conservatively Conservatively Flux must be conserved, minimal dissipation. Flux must be conserved, minimal dissipation. No appearance of unrealistic value. No appearance of unrealistic value. To ensure stability To ensure stability

4. Multi-domain examples From simple to complex From simple to complex

5. Multi-domain in TIMCOM Features Features Both Cartesian grids (TAI and NPB) Both Cartesian grids (TAI and NPB) Same orientation, fixed in space Same orientation, fixed in space NPB is twice the size of TAI NPB is twice the size of TAI Boundary faces of TAI positioned exactly at center of NPB cells Boundary faces of TAI positioned exactly at center of NPB cells

6. Algorithm: TAI to NPB grid UWNPB(JJ,K)= U(I3,J,K) +U(I3,J+1,K) UWNPB(JJ,K)= U(I3,J,K) +U(I3,J+1,K) UWNPB is west face value UWNPB is west face value U(I,J,K) from TAI U(I,J,K) from TAI I3 = I0-3 I3 = I0-3

7. Algorithm: TAI to NPB grid U2NPB(JJ,K)=0.25*[U2(I2,J,K)+ U2(I1,J,K)+ U2NPB(JJ,K)=0.25*[U2(I2,J,K)+ U2(I1,J,K)+ U2(I2,J+1,K)+ U2(I1,J+1,K)] U2(I2,J+1,K)+ U2(I1,J+1,K)] U2NPB is center value U2NPB is center value U2NPB = U@(I=2) U2NPB = U@(I=2) U2(I,J,K) from TAI U2(I,J,K) from TAI I2 = I0-2 I2 = I0-2 Average of 4 values Average of 4 values Same for V, S and T Same for V, S and T Same for U1NPB, Same for U1NPB, except different cell except different cell

8. Using interpolated values in NPB U(1,127,K)=IN(2,127,K)*(FLT1*UWNPB(J, K)+ FLT2*U(1,127,K)) U(1,127,K)=IN(2,127,K)*(FLT1*UWNPB(J, K)+ FLT2*U(1,127,K)) U(1,127,K) is face value U(1,127,K) is face value IN - Mask array for scalar quantities IN(I,J,K)=1,0 for water, land respectively IN - Mask array for scalar quantities IN(I,J,K)=1,0 for water, land respectively UWNPB - Interpolated face value from TAI UWNPB - Interpolated face value from TAI FLT1/2 – time filter to improve stability since TAI has finer grid, smaller time step. Use part of old value to improve stability FLT1/2 – time filter to improve stability since TAI has finer grid, smaller time step. Use part of old value to improve stability

9. Using interpolated values in NPB TMP=U2NPB(J,K)-U1(2,N,K) TMP=U2NPB(J,K)-U1(2,N,K) TMP represents a small incremental difference TMP represents a small incremental difference Same effect as previous case to improve stability Same effect as previous case to improve stability U2(2,N,K)=U2(2,N,K)+TMP U2(2,N,K)=U2(2,N,K)+TMP U2(2,N,K) is center value U2(2,N,K) is center value To give a smaller and smoother increment To give a smaller and smoother increment Improves stability Improves stability Same for V, S and T Same for V, S and T

10. Using interpolated values in NPB U2(1,N,K)=U1NPB(J,K) U2(1,N,K)=U1NPB(J,K) U2(1,N,K) is center value U2(1,N,K) is center value U1NPB - Interpolated center value from TAI U1NPB - Interpolated center value from TAI U1NPB = U@(I=1) U1NPB = U@(I=1) Same for V, S and T Same for V, S and T No time filter due to spatial averaging No time filter due to spatial averaging

11. Algorithm: NPB to TAI grid UETAI(J,K)=U(2,(J-2)/2+127,K) UETAI(J,K)=U(2,(J-2)/2+127,K) UETAI is east face value UETAI is east face value U(I,J,K) from NPB U(I,J,K) from NPB

12. Algorithm: NPB to TAI grid UI0TAI(J,K)=U2(3,(J-2)/2+127,K) UI0TAI(J,K)=U2(3,(J-2)/2+127,K) UI0TAI is center value UI0TAI is center value U2(I,J,K) from NPB U2(I,J,K) from NPB Same for V, S and T Same for V, S and T

13. Using interpolated values in TAI U(I1,J,K)=IN(I1,J,K)*UETAI(J,K) U(I1,J,K)=IN(I1,J,K)*UETAI(J,K) U(I1,J,K) is face value U(I1,J,K) is face value IN - Mask array for scalar quantities IN(I1,J,K)=1,0 for water, land respectively IN - Mask array for scalar quantities IN(I1,J,K)=1,0 for water, land respectively No time filter No time filter U2(I0,J,K)=UI0TAI(J,K) U2(I0,J,K)=UI0TAI(J,K) U2(I0,J,K) is center value U2(I0,J,K) is center value No time filter and IN multiplication required No time filter and IN multiplication required

14. Conclusion Multi-domain problems requires the use of grid coupling Multi-domain problems requires the use of grid coupling Objective to transfer values from one grid to another Objective to transfer values from one grid to another Efficient and conservative algorithm Efficient and conservative algorithm Use of filter to ensure smooth transition and stability Use of filter to ensure smooth transition and stability