On the numerical solutions of the Burgers' type parti̇al di̇fferenti̇al equati̇ons wi̇th hi̇gh order spli̇tti̇ng methods

In this thesis, high order splitting methods have been used for calculating the numerical solutions of the Burgers' type partial differential equations in one space dimension with different boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and can not be used for time-irreversible systems, such as Burgers' type equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients having positive real parts and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem (or can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers' type equations. The numerical results show that the methods with complex 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 extrapolation methods derived from a lower order splitting method produce very accurate solutions of the Burgers' type equations.