TY  - JOUR
A1  - Staniforth, Andrew
A1  - Wood, Nigel
A1  - Reich, Sebastian
T1  - A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations
JF  - Quarterly journal of the Royal Meteorological Society
N2  - A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.
KW  - regularization
KW  - temporal discretization
Y1  - 2006
U6  - https://doi.org/10.1256/qj.06.30
SN  - 0035-9009
VL  - 132
IS  - 621C
SP  - 3107
EP  - 3116
PB  - Wiley
CY  - Weinheim
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - Linearly implicit time stepping methods for numerical weather prediction
JF  - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians
N2  - The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.
KW  - numerical weather prediction
KW  - linearly implicit time stepping methods
KW  - semi-Lagrangian method
KW  - Stormer-Verlet method
KW  - shallow-water equations
Y1  - 2006
U6  - https://doi.org/10.1007/s10543-006-0065-0
SN  - 0006-3835
VL  - 46
SP  - 607
EP  - 616
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Frank, Jason
A1  - Moore, Brian E.
A1  - Reich, Sebastian
T1  - Linear PDEs and numerical methods that preserve a multisymplectic conservation law
N2  - Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]
Y1  - 2006
UR  - http://epubs.siam.org/sisc/
U6  - https://doi.org/10.1137/050628271
SN  - 1064-8275
ER  -