An efficient geometric method for incompressible hydrodynamics on the sphere

We present an efficient and highly scalable geometric numerical method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite -dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable nu-merical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is O(N3) per time-step for N2 spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the paral-lel library ScaLAPACK. Scaling tests show approximately linear scaling up to around 2500 cores for the matrix size N = 4096 with a computational time per time-step of about 0.55 seconds. These results allow for long-time simulations and the gathering of statis-tical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution N = 2048.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

We propose an efficient and scalable numerical method for solving two-dimensional ideal fluid dynamics on the sphere. The method utilizes a tridiagonal splitting of the discrete spherical Laplacian and optimized numerical algorithms. For time-stepping, an isospectral integrator is adopted to preserve the geometric structure of Euler's equations. The algorithm achieves high computational performance by formulating the matrix Lie algebra basis through tridiagonal eigenvalue problems and implementing an efficient parallel computation.