Evaluating Computational Methods under the Influence of Noise Terms

This paper presents a comprehensive analysis of the Kardar-Parisi-Zhang (KPZ) equation's behavior under various noise conditions and investigates the efficiency of different discretization methods, particularly focusing on the exponential decreasing method and Leapfrog-hopscotch methods. By implementing a series of numerical experiments on a standard computing system, this study evaluates the performance of these numerical approaches in terms of computation speed, accuracy, and stability. Our findings reveal that the Leapfrog-hopscotch method, especially in the absence of noise and under Brownian conditions, exhibits significantly faster computation speeds while maintaining high precision in the simulation of stochastic differential equations (SDEs). Furthermore, the sensitivity of simulation outcomes to changes in both nonlinear (λ) and linear (D) parameters of noise are investigated, which offers new insights into the KPZ equation's response to stochastic influences. These results enhance our understanding of the behavior of complex systems and guide the selection of appropriate numerical methods for practical applications in computational physics, mathematical biology, financial modeling, and stochastic control.