LAUSR

In this study, the coupling of the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations are solved that model the motion of shallow-water waves, by a collocation technique based on quintic B-spline as basis functions. For this, the spatial domain is discretized using the quintic B-spline collocation method, which leads to a system of first-order ordinary differential equations. A strong stability preserving Runge–Kutta method of four stages and third-order (SSP-RK43) is applied to solve the obtained system of equations. All the calculations are performed without any linearization or transformation. A couple of test problems are solved to show the efficacy and accuracy of the technique by calculating the $$L_{2}$$ and $$L_{\infty }$$ error norms as well as the discrete energy ( $${\mathcal {E}}$$ ) and mass ( $${\mathcal {Q}}$$ ) conservation properties. This equation is considered because it has great importance in the field of oceanography and the proposed technique is followed due to its ease of implementation and good accuracy. The stability analysis of the technique is performed using the concept of the Jacobian matrix along with eigenvalues and is shown to be stable. Comparison with the existing result shows that the proposed method is more accurate with a higher-order of convergence as compared to many existing techniques.