<p>In the present manuscript, the fifth-order nonlinear Korteweg-de Vries-Kawahara (KdV-Kawahara) equation including two dispersion terms is going to be handled. For the solution process, both finite difference and quadrature differential methods are used. The newly obtained results are important due to two aspects. First, they are better than those of most available ones. The second and most important, the results are found at the low cost of computational efforts. These important aspects ensure that they can also be applied to a wide range of problems encountered in various fields of applied sciences. Moreover, the computed results get closer to the exact ones when the step sizes refine. Travelling single solitary wave and the superposition of two solitary waves solutions are obtained numerically. A comparison of the numerical results shows that present algorithm by contribution of the two numerical methods obtained higher accurate solutions successfully.</p>