A Novel Numerical Simulations for Fornberg-Whitham and Modified Fornberg-Whitham Equations with Nonhomogeneous Boundary Conditions (Journal of Nonlinear Modeling and Analysis, 2024)
In this study, the numerical solutions of the Fornberg-Whitham(FW) equation modeling the qualitative behavior of wave refraction and themodified Fornberg-Whitham (mFW) equation describing the solitary wave andpeakon waves with a discontinuous first derivative at the peak have been obtained. To obtain numerical results, the collocation finite element method hasbeen combined with quintic B-spline bases. Although there are solutions tothese equations by semi-analytical and analytical methods in the literature,there are very few studies using numerical methods. The stability analysis ofthe applied method is examined by the von-Neumann Fourier series method.We have considered four test problems with nonhomogeneous boundary conditions that have analytical solutions to show the performance of the method.The numerical results of the two problems are compared with some studies inthe literature. Additionally, peakon wave solutions and some new numericalresults of the mFW equation, which are not available in the literature, aregiven in the last two problems. No comparison has been made since thereare no numerical results in the literature for the last two problems. The error norms $L_2$ and $L_∞$ are calculated to demonstrate the presented numericalscheme’s accuracy and efficiency. The advantage of the scheme is that it produces accurate and reliable solutions even for modest values of space and timestep lengths, rather than small values that cause excessive data storage in thecomputation process. In general, large step lengths in the space and time directions result in smaller matrices. This means less storage on the computerand results in faster outcomes. In addition, the present method gives moreaccurate results than some methods given in the literature.
Keywords: Fornberg-Whitham Equation; modified Fornberg-Whitham equation; solitary waves; peakon waves; wind waves; quintic B-spline bases; collocation method.