Offenbach Semi-Implicit Time Integration (NH Schemes of the Next Generation) J. Steppeler, DWD

Offenbach Accuracy/high (Spatial) Approximation order Z-Coordinate Efficiency: High Order of Time Integration; Two Time Levels; Serendipidity Grids; “Large” Time Steps; Implicit Time Integration Some Desirable features of Next Generation NH Models

Offenbach Z-Coordinate/3rd Order Spatial Approximation/Serendipidity Grids Different NH Approaches Solution of non-Periodic Problems by Generalised Fourier Transform Efficiency Considerations Computational Example of a non-Periodic Problem Plan of Lecture

Offenbach

Ilder Heinz-Werner 3-D Cloud-Picture 18 January1998 The mountain related bias of convectional clouds and precipitation is supposed to disappear with the Z-coordinate

Offenbach

The geometry of 1 st, 2 nd and 3 rd order representation of the  (x) field 

Offenbach Third Order finite element representation (Continuous cubic spline) Basis functions: 1 - d: Auxiliary grid point Main grid point Field represention: Field operators: All classical grid operations Operations using the polynomial derivatives (inhomogeneous grid operations) Galerkin operations

Offenbach Order consistent field representations in one and three space dimensions Order consistent interpolation requires the field only in the solid lines

Offenbach

The continuous splines in the presence of Orography:  (x) must be a „ smooth“ function (not a step function) =points to pose boundary conditions. The dashed parts of the grid lines are used to pose boundary values.

Offenbach Implicit Approaches

Offenbach The LH - is the Most Common SI Method The Equations of Motion are homogeneously linearised at each Grid Point At Each Grid-Point a Problem of Constant Coefficients is Defined For Each Grid-Point The Associated Linear Problem Can be Solved Using an FT and a Linear Problem Specific to Each Grid-Point The GFT (Generalised FT) Computes the Results of the Different FTs Using One Transform The numerical cost of GFT is Simlar to that of an FT A Fast GFT exists similar to Fast FT Direct Methods for Locally Homogenized SI

Offenbach Split Representation of Derivatives

Offenbach The Fourier Coefficients are the same for the grid Points of a Subregion The Linearised Eqs. are different for each Gridpoint In Case of only One Subregion the Support Points of the derivative are the Boundary Values does not create time-Step Limitations GFT Returns the Grid-Point-Values after doing a Different Eigenvalue Calculation at Each Point Organisation of the Implicit Time-Step

Offenbach The SI Timestep Subtract Large Scale Part of each function Compute Fourier Coefficients Choose Gridpoint Compute next time level in Fourier Space Transform Back Use Result Only for Chosen Grid-Point Do the Above for all Other Grid Points Set Boundary Values as in Finite Difference Methods

Offenbach Example:1-d Schallow Water Eq Definitions: SI Scheme: Periodicity Terms (=0in the following)

Offenbach Boundary- and Exterior Points Point to pose (Artificial) Boundary Values Redundant Points Redundant points can be included in the FT The result of the time-step does not depend on the continuation of the field to redundant points

Offenbach Operation in Fourier Space Definition: SI Scheme: Linear Equations at the chosen gridpoint:

Offenbach Computational Example: 1-d Schock Wave The time Step could be increased up to the CFL of advection (10 m/sec)

Offenbach A direct si- method was proposed The method is based on a generalised Fourier Transform The generalised FT is potentially as efficient as the FT (fast FT) The method is efficient for increased spatial order 1-d tests have been performed Conclusions