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In this work, high order splitting methods have been used for calculating thenumerical solutions of the Burgers' equation in one space dimension withperiodic and Dirichlet boundary conditions. However, splitting methods withreal coefficients of order higher than two necessarily have negativecoefficients and can not be used for time-irreversible systems, such as Burgersequations, due to the time-irreversibility of the Laplacian operator.Therefore, the splitting methods with complex coefficients and extrapolationmethods with real and positive coefficients have been employed. If we considerthe system as the perturbation of an exactly solvable problem(or can be easilyapproximated numerically), it is possible to employ highly efficient methods toapproximate Burgers' equation. The numerical results show that the methods withcomplex time steps having one set of coefficients real and positive, say$a_i\in\mathbb{R}^+$ and $b_i\in\mathbb{C}^+$, and high order extrapolationmethods derived from a lower order splitting method produce very accuratesolutions of the Burgers' equation.

Numerical solution of the Burgers' equation with high order splitting methods