Stability of Tension B-Spline-Based Differential Quadrature Approaches for Nonlinear Klein-Gordon and Sine-Gordon Equations in Engineering Systems

Source abstract

Accurate and computationally efficient simulation of nonlinear Klein-Gordon and Sine-Gordon equations is important for analyzing nonlinear wave propagation, oscillatory dynamics and localized structures arising in engineering and physical systems. In this work, two differential quadrature approaches with cubic Uniformly Algebraic Trigonometric (UAT) and Uniformly Algebraic Hyperbolic (UAH) tension B-splines are proposed and compared. The spatial derivatives are discretized by using the corresponding spline-based DQM weighting coefficients. The resulting semi-discrete systems are integrated in time by the Runge-Kutta 4(3) scheme. The methods are tested on the one-dimensional Klein-Gordon and the one- and two-dimensional Sine-Gordon benchmarks. Accuracy is assessed through L ∞ , RMS, and relative errors, exact-numerical solution comparisons, grid- and time-dependent error behavior, and convergence analysis. Matrix/eigenvalue analysis is employed to investigate numerical stability, while computational complexity, CPU time, wall-clock time, and memory requirements are evaluated to assess efficiency. The UAT- and UAH-based formulations produce nearly identical numerical results under matched computational conditions. Representative errors remain bounded, ranging from approximately 10 –9 – 10 –10 for highly accurate Klein–Gordon cases 10 –4 – 10 –5 to for the considered Sine-Gordon benchmarks. The stability and computational analyses further demonstrate reliable performance with modest memory requirements. These results establish both tension B-spline DQM formulations as accurate and computationally viable alternatives for nonlinear wave problems in scientific and engineering applications.