Malaysian Journal of Mathematical Sciences, September 2026, Vol. 20, No. 3
On Fourth-Order Staggered Compact Finite Difference–Runge–Kutta Methods for Shallow Water Equations
Mohamad Azli, P. N. F., Loy, K. C., Mohd Mahali, S., and Abdullah, H. H.
Corresponding Email: kakchoon@umt.edu.my
Received date: 9 September 2025
Accepted date: 22 January 2026
Abstract:
This paper introduces a high-order numerical method for solving the one-dimensional shallow water equations (1D SWEs), a simplified form of the Navier–Stokes equations widely used for long-wave propagation studies. The method combines a fourth-order staggered compact finite difference scheme in space with two time integrators: the classical fourth-order Runge–Kutta method (RK4) and its strong stability preserving variant (SSP–RK(5,4)). Through semi-analytical benchmark problems involving Gaussian, solitary, and Cauchy wave profiles, we verify that the scheme consistently achieves global fourth-order accuracy, remains CPU-efficient, and preserves smooth wave propagation with high fidelity. Comparisons with analytical datasets (Synolakis, 1986/1987) further confirm satisfactory agreement for wave elevation and maximum amplitude in shoaling scenarios. Although the present approach does not enforce strict conservation, well-balancing, or positivity-preserving properties, it nevertheless shows that compact schemes employing lower-order stencils can deliver spectral-like convergence and serve as a viable alternative to high-order finite-difference solvers during smooth propagation and shoaling phases. These results highlight the potential of high-order staggered compact schemes as accurate and computationally efficient tools for wave modelling, with future extensions aimed at incorporating well-balancing, wet-dry treatments, and two-dimensional applications.
Keywords: shallow water equations; high-order methods; fourth-order accuracy; staggered compact scheme; Runge–Kutta methods; tsunami propagation; wave shoaling.
