Abstract
This paper has employed a comparative study between the numerical scheme and stability condition. Numerical calculations are carried out based on three different numerical schemes, namely the central finite difference, fourier leap-frog, and fourier spectral RK4 schemes. Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme.
Highlights
The Korteweg-de Vries (KdV) equation ut + uux + uxxx = 0, (1)is a nonlinear, dispersive partial differential equation for a function where u (x, t) of two real variables, space x and time t
Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme
Three numerical schemes including the finite difference, fourier leap-frog and fourier RK4 procedures are presented for the equation
Summary
Is a nonlinear, dispersive partial differential equation for a function where u (x, t) of two real variables, space x and time t. It is a mathematical model of waves on shallow water surfaces and notable as the prototypical example of an exactly solvable model, i.e., a non-linear partial differential equation whose solutions can be exactly and precisely specified. The study of non-linear waves would not have been so successful had it not done with stable numerical schemes, especially for time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded. We analyze the stability of three numerical schemes on the KdV equation based on von Neumann method and conclude that Fourier RK4 scheme can meet the stability criterion with suitable spatial and time steps
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