Advanced Numerical Methods

The goal of this session is to provide an overview of the use of generalised curvilinear coordinates in atmospheric numerical models.

By the end of the session you should be able to:

• describe some important aspects of the formulation and implementation of the governing equations in generalised coordinates

• describe various vertical coordinates employed in atmospheric models

• indicate the use of generalised coordinates to employ moving mesh adaptivity

The success of the spectral transform method in global NWP in comparison to alternative methods has been overwhelming, with many operational forecast centres (including ECMWF) having madethe spectral transform their method of choice. The lecture will introduce the basic elements of the spectral transform, explain why it has been successful and describe recent developments such as
the fast Legendre transform.

By the end of the session you should be able to:

•   explain what the spectral transform method is, how it is applied, and describe the latest developments at ECMWF.

•   give reasons why it is successful for global NWP and climate.

•   identify potential disadvantages of the method.

The aim of this set of lectures is to systematically build theoretical foundations for Numerical Weather Prediction at nonhydrostatic resolutions. In the first part of the lecture, we will discuss a suite of all-scale nonhydrostatic PDEs, including the anelastic, the pseudo-incompressible and the fully compressible Euler equations of atmospheric dynamics. First we will introduce the three sets of nonhydrostatic governing equations written in a physically intuitive Cartesian vector form, in abstraction from the model geometry and the coordinate frame adopted. Then, we will combine the three sets into a single set recast in a form of the conservation laws consistent with the problem geometry and the unified solution procedure. In the second part of the lecture, we will build and document the common numerical algorithm for integrating the generalised set of the governing PDEs put forward in the first part of the lecture. Then, we will compare soundproof and compressible solutions and demonstrate the efficacy of this unified numerical framework for two idealised flow problems relevant to weather and climate.

By the end of the lectures you should be able to:

• explain the form, properties and role of alternative systems of nonhydrostatic PDEs for all scale atmospheric dynamics;

• explain the importance and key aspects of continuous mappings employed in all-scale atmospheric models;

• explain the difference between the explicit and semi-implicit algorithms for integrating nonhydrostatic PDEs, the importance of consistent numerical approximations, and the fundamental role of transport and elliptic solvers.

Using the 30-year history of ECMWF's Integrated Forecasting System (IFS) as an example, thelecture is an introduction to the development and current state-of-the-art of global numerical weather prediction (NWP), as well as to the challenges faced in the future. It is intended to provide
an overview and context for the topics covered in more detail during the course.

By the end of the session you should be able to:

•   describe the development of global NWP, the current-state-of-the-art, and future challenges

•   identify relevant areas of research in numerical methods for Earth-System Modelling

•   put into context every subsequent lecture and its purpose

During this presentation, we will discuss two of the questions faced by numerical weather prediction scientists as forecast models reach horizontal resolutions of 6 to 2 km:

• Do we need to abandon the primitive equations for a non-hydrostatic system of equations?

• Do we still need a deep convection parametrisation?

• and we will show what answers to these questions are given by very high resolution simulations of the IFS.

By the end of the presentation, you should be able to:

• discuss the limits of the hydrostatic approximation for numerical weather prediction

• explain the dilemma of parametrizing deep convection versus permitting explicit deep convection at resolution in the grey zone of convection

The aim of this session is to understand the main issues and challenges in parallel computing, and how parallel computers are programmed today.

By the end of this session you should be able to

• explain the difference between shared and distributed memory

• describe the key architectural features of a supercomputer

• describe the purpose of OpenMP and MPI on today’s supercomputers

• identify the reasons for the use of accelerator technology

The aim of two lectures is to introduce basis of finite volume and continuous finite element discretisations and relate them to corresponding data structures and mesh generation techniques. The main focus will be on unstructured meshes and their application to global and local atmospheric models. Flexibility, communication overheads, memory requirements and user friendliness of such meshes with be contrasted with those of structured meshes. The most commonly used mesh generation techniques will be highlighted, together with mesh manipulation techniques employed in mesh adaption approaches and will be followed by a discussion of alternative geometrical representations of orography. An example of unstructured meshes’ implementation to non-hydrostatic and hydrostatic atmospheric solvers will provide an illustration of their potential and challenges.

By the end of the lecture you should be able to:

• understand applicability, advantages and disadvantages of selected mesh generation techniques for a given type of application.

• appreciate importance of data structures in relation to atmospheric models and mesh generation.

• gain awareness of issues related to flexible mesh generation and adaption.

The aim of this session is to understand how numerical precision can be traded against computational performance in Earth System modelling. It will be discussed how a reduction in numerical precision will influence model quality and how the minimal level of precision that will still allow simulations at high accuracy can be identified. We will give an overview about existing hardware options to adjust numerical precision to the need of the application.

By the end of this session you should be able to

• describe how rounding errors will impact model simulations that show chaotic dynamics

• describe the connection between numerical precision, computational performance and predictability
• recall how a trade off between precision and performance can be realised in Earth System modelling today and in the future

The aim of two lectures is to introduce basis of finite volume and continuous finite element discretisations and relate them to corresponding data structures and mesh generation techniques. The main focus will be on unstructured meshes and their application to global and local atmospheric models. Flexibility, communication overheads, memory requirements and user friendliness of such meshes with be contrasted with those of structured meshes. The most commonly used mesh generation techniques will be highlighted, together with mesh manipulation techniques employed in mesh adaption approaches and will be followed by a discussion of alternative geometrical representations of orography. An example of unstructured meshes’ implementation to non-hydrostatic and hydrostatic atmospheric solvers will provide an illustration of their potential and challenges.

By the end of the lecture you should be able to:

• understand applicability, advantages and disadvantages of selected mesh generation techniques for a given type of application.

• appreciate importance of data structures in relation to atmospheric models and mesh generation.

• gain awareness of issues related to flexible mesh generation and adaption.

The aim of this session is to describe Eulerian rather than Lagrangian type numerical techniques for integrating the equation sets encountered in NWP models. We will present an overview of different time-stepping techniques and discuss the advantages and disadvantages of each approach.

By the end of the session you should be able to:

• obtain a good understanding of the minimum theoretical properties required by time-stepping schemes
   
• describe differences, strengths-weaknesses of different time-stepping approaches such as split-explicit time-stepping, Runge-Kutta time-stepping
  

• describe the basic features of different time-stepping schemes used in other weather forecasting models such as WRF, ICON
  

The aim of this session is to learn about recent developments in discontinuous higher order spatial discretization methods, such as the Discontinuous Galerkin method (DG), and the Spectral Difference method (SD). These methods are of interest because they can be used on unstructured meshes and facilitate optimal parallel efficiency. We will present an overview of higher order grid point methods for discretizing partial differential equations (PDE's) with compact stencil support, and illustrate a practical implementation.

By the end of the session you should be able to:

• ell what are the advantages offered by discontinuous higher order methods

• describe how to solve PDE's with discontinuous methods

• identify the key elements that contribute to a PDE solver