Spectral methods for solving partial differential equations in multiphysics simulations

Spectral methods approximate partial differential equations (PDEs) by expressing the solution as a sum of globally defined basis functions. In multiphysics simulations such as coupled fluid flow and heat transfer, spectral discretisations offer high accuracy for smooth solutions and avoid the numerical diffusion associated with low order schemes. Challenges include choosing appropriate basis functions for complex domains, dealing with nonlinear terms and ensuring numerical stability when the equations contain multiple spatial and temporal scales. This paper formulates the governing equations for an incompressible viscous fluid coupled with a heat transport equation, expands the variables in Fourier and Chebyshev bases and derives a semi discrete system using a collocation strategy. We demonstrate exponential convergence in space for smooth solutions and compare spectral accuracy with finite difference accuracy on a benchmark problem. The results show that spectral methods achieve a given error with significantly fewer degrees of freedom, leading to reduced computational cost for moderate problem sizes. Outcomes include guidelines for choosing spectral bases in multiphysics contexts and identification of stability constraints arising from nonlinear convective terms.