Accuracy and convergence of finite difference and Fourier spectral methods for a nonlinear fluids-dynamics PDEs

                @article{CRSIJ26000410,
                  author = {Makinta Bakura , Ahmadu M. Brono, Usman A. Marte, Abba V. Mandara, Abubakar Masha, Laminu Dahiru},
                  title = {Accuracy and convergence of finite difference and Fourier spectral methods for a nonlinear fluids-dynamics PDEs },
                  journal = {Cosmo Research and Science International Journal},
                  year = {2026},
                  volume = {2},
                  number = {2},
                  pages = {597-613},
                  issn = {3108-1584},
                  url = {https://cosmorsij.com/published/CRSIJ26000410.pdf},
                  abstract = {Nonlinear partial differential equations (PDEs) play a crucial role in modelling various processes like transport, diffusion, and fluid dynamics. To solve these equations reliably, we need discretization methods that we can trust in terms of their accuracy, stability, and convergence. In this study, we dive into a comparative numerical analysis of two methods: a second-order central finite-difference (FD2) method and a Fourier spectral method, specifically applied to the one-dimensional viscous Burgers equation. We tackle this problem within a periodic domain, starting with a smooth sinusoidal initial condition and a viscosity coefficient. To create a high-accuracy reference solution, we use the Cole–Hopf transformation, which simplifies the nonlinear Burgers equation into a linear heat equation. Both methods utilize the classical fourth-order Runge–Kutta technique for time integration. We conduct systematic spatial refinement experiments for the FD2 method with grid points set at 32, 64, 128, and 256, and for the Fourier spectral method with 16, 32, 64, and 128 Fourier modes. We assess numerical accuracy through error norms and calculate the observed convergence rates from successive refinements. The results from the finite-difference method show an approximate second-order convergence, aligning with the expected accuracy of the spatial discretization. In contrast, the Fourier spectral method demonstrates a much faster reduction in error for the smooth periodic benchmark, which is in line with the spectral approximation properties known for smooth solutions. Overall, our findings indicate that while FD2 offers a straightforward and reliable numerical baseline, the Fourier spectral method shines when the solution is smooth and the geometry and boundary conditions are well-suited for a global Fourier representation. This study highlights the significance of having independent reference solutions, systematic refinement, and reproducible computational procedures when verifying numerical methods for nonlinear PDEs.},
                  keywords = {Burgers equation, finite difference and Fourier spectral method, nonlinear PDEs, convergence analysis, Cole–Hopf transformation, numerical verification and fluid dynamics},
                  month = {September}
        }