October 26, 2017 — Support my next blog post, buy me a coffee ☕.
My latest paper, with my former colleague Yuji Nakatsukasa, has just been accepted in SISC. In this post, I review the main ideas of our work.
We are interested in computing smooth solutions of stiff PDEs on the unit sphere of the form $$ u_t = \alpha\Delta u + \mathcal{N}(u), \;\; u(0,x,y,z)=u_0(x,y,z), $$ where \(u(t,x,y,z)\) is a function of time \(t\) and Cartesian coordinates \((x,y,z)\) with \(x^2 + y^2 + z^2=1\). The function \(u\) can be real or complex and the PDE can be a single equation, as well as a system of equations. A large number of PDEs of interest in science and engineering take this form. Examples include the Allen–Cahn equation \(u_t = \epsilon\Delta u + u - u^3\), the nonlinear Schrödinger equation \(u_t=i\Delta u + iu|u|^2\), the Ginzburg–Landau equation (the picture at the top is a solution), and all reaction-diffusion equations.
Our algorithms are based on a variant of the double Fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones, and implicit-explicit time-stepping schemes. Operating in coefficient space with these new matrices allows one to use a sparse direct solver, avoids the coordinate singularity, and maintains smoothness at the poles, while implicit-explicit schemes circumvent severe restrictions on the time-steps due to stiffness. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform best. Implementations in MATLAB and Chebfun make it possible to compute the solution of many PDEs to high accuracy in a very convenient fashion—check out the spinsphere code.
I hope to use this code for investigating pattern formation on the sphere with Philippe Trinh, Michael Ward, and Stanislav Shvartsman. I'm visiting Stanislav at Princeton in a couple of weeks—looking forward to it!
Blog posts about spectral methods
2020 Exponential integrators for stiff PDEs
2018 Computer-assisted proofs for PDEs
2018 Spherical caps in cell polarization
2018 Solving nonlocal equations on the sphere
2018 Gibbs phenomenon and Cesàro mean
2017 Solving PDEs on the sphere
2017 When planets dance