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Highlights • Space and laboratory plasma supports Maxwellian and non-Maxwellian distributions. • Efficient approach presented to incorporate both distributions in fluid simulation. • Successive over relaxation method improves the performance of the Poisson solver. • Methodology to determine optimized value of the relaxation parameter is presented. • New approach is tested for the fluid simulation of ion acoustic waves in plasma. Abstract Solitary waves are commonly observed in both thermal and nonthermal plasmas. Thermal plasmas are generally represented by the Maxwellian distribution, whereas the nonthermal plasmas represented by the non-Maxwellian distributions, such as Kappa or Cairns distributions. Numerous theoretical fluid models have used these distributions to model the waves in space and laboratory plasmas. However, apart from our recent studies, there are no attempts to include these distributions in a fluid simulation of waves in plasma. In this paper, we present an efficient approach to deal with the stability and convergence of the Poisson solver in the fluid simulation of plasma that follows the Kappa, Maxwell, and Cairns distributions. We used the stationary iterative methods, namely, Jacobi (JA), Gauss–Seidel (GS) and Successive Over Relaxation (SOR) in the development of the Poisson solver. Our results show that the SOR method significantly improves the performance towards the stability and convergence of the Poisson solver for the plasma with all three-velocity distributions. The new fluid code with SOR Poisson solver is applied to examine the evolutionary characteristics of the ion acoustic solitary waves in a plasma and they are found to be in agreement with the fluid theory. [ABSTRACT FROM AUTHOR]