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Abstract

This paper investigates a numerical strategy for the one-dimensional Burgers equation with a nonzero source term. Such equations arise in simplified models of transport and diffusion processes and are often used to assess the performance of numerical schemes for nonlinear evolution problems. The proposed approach combines a second-order Crank–Nicolson time discretization with a projection-based procedure that separates the nonlinear convective contribution from diffusive effects. Spatial approximation is carried out using a Chebyshev spectral collocation method, which provides high accuracy for smooth solutions with a limited number of degrees of freedom. The resulting fully discretized system is solved through an iterative procedure designed to handle the nonlinearity efficiently, while a spectral filtering step is employed to control spurious high-frequency oscillations. Numerical experiments are performed using a manufactured smooth solution in order to evaluate accuracy and convergence with respect to both time step and spatial resolution. The results demonstrate that the proposed method produces stable and accurate approximations and exhibits the expected convergence behavior. Comparisons with representative numerical results reported in the literature further support the validity of the approach. The study shows that the proposed framework offers a reliable and effective tool for solving Burgers-type equations with external forcing.

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