- Revise ESSPRK paper based on reviewers comments and additional resutls.
- Prepare research proposal on Downwind methods.
- ESSPRK paper (with David, Colin and Jim)
Comments:
- Include additional new results about the family of effective order four, classical order two ESSPRK methods with \(s^2+1\) stages.
- Read papers about Multirate methods.
- Numerical implementation of 1st order Multirate schemes on advection equation.
- Numerical implementation of 1st order Multirate schemes on advection equation. (cont.)
- Change research proposal topic: Focus on Downwind schemes.
- Read “Step Sizes for Strong Stability Preservation with Downwind-Biased Operators” (Ketcheson D.) and relative chapter in SSP book.
- Search for implicit DWSSPRK(2,2) with arbitary large SSP coeffcient - prepare a Maple script.
- Reformulate optimization of ESSPRK methods based on coefficients of alpha matrix in the Shu-Osher form.
- Find a general form for ESSPRK(\(s^2+1,4,2\)) with \(C=s^2-s\).
- September 29 - October 3:
- Find algebraically the coefficients of ESSPRK(\(s^2+1,4,2\)) with \(C=s^2-s\).
- Literature review for DW methods.
- Search for possible implicit DWSSPRK(2,2) and DWSSPRK(s,3) with arbitary large SSP coeffcient. (use mathematica)
- Answer questions posed in “Step Sizes for Strong Stability Preservation with Downwind-Biased Operators” and relative David’s wiki comments.
Ideas: 1. Fix a number of parameters and plot the polynomial as a function of the remaining parameters. Examine the behavior of the polynomial for various values of a.m.r. 2. Work on the stability function of DW methods. Examine the relation of the order conditions of the stability function and SSP conditions.
- Further reading about DW methods.
- Reformulate optimization for starting and finishing methods based on the Shu-Osher form - make code faster for large number of stages.
- Search for a sparce pattern of relative starting and finishing methods for ESSPRK(\(s^2+1,4,2\)).
- Merge code of ESSPRK in RK-opt.
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