| 1. |
|
|
| 2. |
- Starius, Göran, 1944
(författare)
-
A Solution to the Pole Problem for the Shallow Water Equations on a Sphere
- 2014
-
Ingår i: TWMS Journal of Pure and Applied Mathematics. - 2076-2585. ; 5:2, s. 152-170
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a reduced gridding technique for the shallow water equations on a sphere, based on spherical coordinates. In a small vicinity of the poles, a longitudinal derivative is discretized at a grid-point on a parallel, by using points on the great circle through the grid-point and tangent to the parallel. Centered one-dimensional interpolation formulas are used in this process and also in connecting adjacent segments in the reduced grid. The remaining spatial discretization is obtained by simply replacing derivatives by centered equidistant finite difference approximations. Numerical experiments for scalar advection equations and for the well-known Rossby-Haurwitz test example indicate that the methods developed work surprisingly well. Some advantages are that (i) a fairly uniform grid, with many reductions or segments, can be used, (ii) order of approximation $2p$ in the spatial discretizations requires only $4p+1$ points and (iii) the local and simple structure of the schemes will make efficient implementation on massively parallel computer systems possible. The paper is an attempt towards global numerical weather prediction models, by first analyzing the pole problem for reduced latitude-longitude grids.
|
|
| 3. |
- Starius, Göran
(författare)
-
Asymptotic expansions for a class of elliptic difference schemes
- 1981
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 37:156, s. 321-326
-
Tidskriftsartikel (refereegranskat)abstract
- This paper derives an asymptotic expansion of the global error for Kreiss' difference scheme for the Dirichlet problem for Poisson's equation. This scheme, combined with a deferred correction procedure or the Richardson extrapolation technique, yields a method of accuracy at least O(h6.5) in L2, where h is the mesh length.
|
|
| 4. |
- Starius, Göran Christer, 1944
(författare)
-
A Solution to the Pole Problem for the Shallow Water Equations on a Sphere
- 2014
-
Ingår i: TWMS Journal of Pure and Applied Mathematics. - 2076-2585. ; 5:2, s. 152-170
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a reduced gridding technique for the shallow waterequations on a sphere, based on spherical coordinates. In a small vicinityof the poles, a longitudinal derivative is discretized at a grid-pointon a parallel, by using points on the great circle through the grid-pointand tangent to the parallel. Centered one-dimensional interpolationformulas are used in this process and also in connecting adjacentsegments in the reduced grid. The remaining spatial discretizationis obtained by simply replacing derivatives by centered equidistantfinite difference approximations.Numerical experiments for scalar advection equations and for thewell-known Rossby-Haurwitz test example indicate that themethods developed work surprisingly well. Some advantagesare that (i) a fairly uniform grid, with many reductions or segments,can be used, (ii) order of approximation $2p$ in the spatial discretizationsrequires only $4p+1$ points and (iii) the local and simple structureof the schemes will make efficient implementation on massivelyparallel computer systems possible.The paper is an attempt towards global numerical weatherprediction models, by first analyzing the pole problem for reducedlatitude-longitude grids.
|
|
| 5. |
- Starius, Göran Christer, 1944
(författare)
-
Fourth Order Partitioned Methods Designed for the Time Integration of Atmospheric Models
- 2023
-
Ingår i: Tellus Series a-Dynamic Meteorology and Oceanography. - 0280-6495 .- 1600-0870. ; 75:1, s. 172-178
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive a fourth order two step Runge-Kutta method with four stages, for additively partitioned systems of ordinary differential equations. Our main objective is that it will be useful as a horizontally explicit and vertically implicit (HEVI) method for atmospheric models. In our method the diagonal coefficients in the implicit part are all equal, and the HEVI-stability properties seem to be excellent. Further, the accuracies obtained for two simple test problems, used with different resolutions and integration intervals, considerably surpass those of a third order one step Runge-Kutta method also with four stages, used for comparison.
|
|
| 6. |
- Starius, Göran Christer, 1944
(författare)
-
On the use of reduced grids in conjunction with the Equator-Pole grid system
- 2020
-
Ingår i: Tellus, Series A: Dynamic Meteorology and Oceanography. - : Stockholm University Press. - 0280-6495 .- 1600-0870. ; 72:1, s. 1-12
-
Tidskriftsartikel (refereegranskat)abstract
- This is a continuation of a previous paper by Starius, where for the solution of the shallow water equations on the sphere, we consider Equator-Pole grid systems consisting of one latitude-longitude grid covering an annular band around the equator, and two orthogonal polar grids based on modified stereographic coordinates. Here, we generalise this by letting an equatorial band be covered with a reduced grid system, which can decrease the total number of grid points by at least 20%, with no substantial change in accuracy. Centred finite differences of high order are used for the spatial discretisation of the underlying differential equations and the explicit fourth order Runge-Kutta method for the integration in time. In the paper mentioned above, we demonstrate accuracy in the total mass, which is much higher than needed for NWP, for smooth solutions. Here we show that this holds also for non-smooth solutions, by considering the Cosine bell no. 1 and the mountain problem no. 5 in Williamson et al., the solutions of which have discontinuous second and first order derivatives, respectively.
|
|
| 7. |
- Starius, Göran Christer, 1944
(författare)
-
"The Equator-Pole grid system': an overset grid system for the sphere, with an optimal uniformity property
- 2019
-
Ingår i: Tellus Series a-Dynamic Meteorology and Oceanography. - : Stockholm University Press. - 1600-0870 .- 0280-6495. ; 70:1
-
Tidskriftsartikel (refereegranskat)abstract
- This article proposes a new grid system for the sphere, which consists of three orthogonal and almost uniform grids. The basic one is a latitude-longitude grid covering an annular band around the equator. For the rest of the sphere this grid is complemented by two grids covering the polar regions, and based on suitably modified stereographic coordinates. The rectangular structure of the grids makes highly efficient implementation on massively parallel computer systems possible. Numerical experiments with the new grid system are carried out for two advection examples, namely smooth deformational flow and rotation of the Cosine bell, and for the test problems 2, 3, and 6 from Williamson etal., concerning the non-linear shallow water equations. For problem 6, the Rossby-Haurwitz wave, we study conservation properties for mass and energy. The computational results compare favourably with results for other grids. Our focus is the definition of the grid system together with the connection between grids by overlapping and the use of centred interpolation formulas. The method of centred finite differences is used for the spatial discretisation of the differential equations, because it is well-known and easy to implement.
|
|
| 8. |
|
|
| 9. |
|
|
| 10. |
|
|
