Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

Effective order strong stability preserving Runge–Kutta methods

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of effective order methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods—like classical order five methods—require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge–Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice

Strong stability preserving explicit Runge-Kutta methods of maximal effective order

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods - like classical order five methods - require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.Comment: 17 pages, 3 figures, 8 table

Numerical representation of mountains in atmospheric models

Numerical weather and climate models are using increasingly fine meshes that resolve small-scale, steeply-sloping terrain. Terrain-following meshes become highly distorted above such steep slopes, degrading the numerical balance between the pressure gradient and gravity. Furthermore, existing models often prefer dimensionally-split transport schemes for their computational efficiency, but such schemes can suffer from splitting errors above steep slopes. The cut cell method offers an alternative that avoids most mesh distortions, but arbitrarily small cut cells can impose severe time-step constraints on explicit transport schemes. This thesis makes three contributions to improve atmospheric simulations, particularly in the vicinity of steeply-sloping terrain. First, a multidimensional finite volume transport scheme is formulated to obtain accurate solutions on arbitrary, highly-distorted meshes. Stability conditions derived from a von Neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage. The scheme achieves second-order convergence across a series of tests using highly-distorted terrain-following meshes and cut cell meshes. The scheme is extended using the k-exact method to achieve third-order convergence on distorted meshes without increasing the computational cost during integration. Second, a new type of mesh is designed to avoid severe mesh distortions associated with terrain-following meshes and avoids severe time-step constraints associated with cut cells. Numerical experiments compare the new mesh with terrain-following and cut cell meshes, revealing that the new mesh simultaneously achieves an accurate balance between the pressure gradient and gravity, and avoids severe time-step constraints. Third, a new two-dimensional test case is proposed that excites the Lorenz computational mode. The new test is used to compare results from a nonhydrostatic model with Lorenz staggering with those from a model variant with a newly-developed generalised Charney–Phillips staggering for arbitrary meshes