Computers and Fluids 157 (2017) 196–207 Contents lists available at ScienceDirect Computers and Fluids journal homepage: www.elsevier.com/locate/compfuid Well-balanced methods for the shallow water equations in spherical coordinates  Manuel J. Castro ∗ , Sergio Ortega, Carlos Parés University of Málaga, Dpto. Análisis Matemático, E.I.O y Matemática Aplicada, Campus de Teatinos S/N, 29071 Málaga, Spain a r t i c l e i n f o Article history: Received 7 March 2017 Revised 10 August 2017 Accepted 27 August 2017 Keywords: Shallow water model Well-balanced methods Finite volume methods Approximate Riemann solvers High order methods, a b s t r a c t The goal of this work is to obtain a family of explicit high order well-balanced methods for the shallow water equations in spherical coordinates. Application of shallow water models to large scale problems re- quires the use of spherical coordinates: this is the case, for instance, of the simulation of the propagation of a Tsunami wave through the ocean. Although the PDE system is similar to the shallow water equations in Cartesian coordinates, new source terms appear. As a consequence, the derivation of high order numer- ical methods that preserve water at rest solutions is not as straightforward as in that case. Finite Volume methods are considered based on a first order path-conservative scheme and high order reconstruction operators. Numerical methods based on these ingredients have been successfully applied previously to the nonlinear SWEs in Cartesian coordinates. Some numerical tests to check the well-balancing and high order properties of the scheme, as well as its ability to simulate planetary waves or tsunami waves over realistic bathymetries are presented. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction The shallow water equations (SWEs) are useful to model free surface gravity waves whose wavelength is much larger than the characteristic bottom depth: see [39–41] for a review of these equations. This is the case of tsunami waves: although the depth bottom of the oceans cannot be considered as small, the character- istic wavelength of a tsunami can be of the order of 100 km, what is significantly larger than the characteristic ocean depth. On the other hand, the application of SWE to large scale phe- nomena (of the order of 1000s of km) makes necessary to take into account the curvature of the Earth. Usually, the Earth is ap- proached by a sphere and the equations are written in spherical coordinates. Although the PDE system is similar to the SWEs in the plane using Cartesian coordinates, new source terms appear due to the change of variables. Therefore, the discretization of the system in spherical coordinates goes far beyond a simple adaptation of the numerical methods for the equations written in Cartesian coordi- nates. SWEs in spherical coordinates are the basis of many of the most used software packages for tsunami simulations. In most cases, the linear SWEs are considered, what is enough to give an ac-  Dedicated to Tito Toro, maestro y amigo, on the occasion of his 70th anniver- sary. ∗ Corresponding author. E-mail address: castro@anamat.cie.uma.es (M.J. Castro). ceptable simulation of the propagation of the wave in deep wa- ters: [2,32]. Nevertheless, the linear SWEs cannot be used for the simulation of the arrival of the wave to the shore and the subse- quent flooding. On the other hand, when the nonlinear SWEs are considered, in most cases the formulation in primitive variables (i.e. velocity/thickness) is used instead of the conserved ones (dis- charge/thickness): see [31]. While the systems written in one or another set of variables is equivalent for smooth solutions, this is not the case when shock waves develop: the jump condition de- pends on the formulation, and the one consistent with the physics of the system is the one corresponding to the formulation in con- served variables. Again, while the formation of shock waves is not expected during the propagation of the wave, it is very likely to happen when the wave is close to the shore. Finally, in some cases the nonlinear SWEs in spherical coordi- nates using the conserved variables is used, but some of the source terms due to the change of variables are neglected, as their influ- ence is not relevant far enough of the poles (see [32]). In this article, we consider the nonlinear SWEs in conserved variables formulation with all the source terms related to bottom variations and to the curvature. Neither Coriolis force (whose influ- ence is not relevant for Tsunami waves) nor friction forces (whose numerical treatment can be done like in the Cartesian coordinates case) are considered. Our goal is to derive an explicit high order well-balanced nu- merical scheme. By well-balanced, we mean that stationary so- lutions corresponding to water at rest situations have to be pre- http://dx.doi.org/10.1016/j.compfluid.2017.08.035 0045-7930/© 2017 Elsevier Ltd. All rights reserved.