We propose a simple and efficient class of direct solvers for Poisson equation in finite or infinite domains related to spherical geometry. The solver was developed based on truncated spherical harmonics expansion, where the differential mode equations were solved by second-order finite difference method without handling coordinate singularities. The solver was further extended to study the dynamics of a diffusiophoretic particle suspended in Stokes flow. Numerical experiments suggested that the particle can achieve a self-sustained unidirectional motion at moderate Péclet numbers, whereas the particle motion becomes chaotic in high Péclet number regimes. The statistical analysis illustrates the run-and-tumble-like nature at short times and diffusive nature at long times without any source of noise.