Taras I. Lakoba

Taras I. Lakoba

We analyze a numerical instability that occurs in the well-known split-step Fourier method on the background of a soliton. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finite-difference schemes. % on the background of a monochromatic wave, considered earlier in the literature. Moreover, the principle of ``frozen coefficients", in which variable coefficients are treated as ``locally constant" for the purpose of stability analysis, is strongly violated for the instability of the split-step method on the soliton background. Our analysis explains all these features. It is enabled by the fact that the period of oscillations of the unstable Fourier modes is much smaller than the width of the soliton.

Taras I. Lakoba, Zihao Deng

We study numerical (in)stability of the Method of characteristics (MoC) applied to a system of non-dissipative hyperbolic partial differential equations (PDEs) with periodic boundary conditions. We consider three different solvers along the characteristics: simple Euler (SE), modified Euler (ME), and Leap-frog (LF). The two former solvers are well known to exhibit a mild, but unconditional, numerical instability for non-dissipative ordinary differential equations (ODEs). They are found to have a similar (or stronger, for the MoC-ME) instability when applied to non-dissipative PDEs. On the other hand, the LF solver is known to be stable when applied to non-dissipative ODEs. However, when applied to non-dissipative PDEs within the MoC framework, it was found to have by far the strongest instability among all three solvers. We also comment on the use of the fourth-order Runge--Kutta solver within the MoC framework.

Taras I. Lakoba, Zihao Deng

We show that imposition of non-periodic, in place of periodic, boundary conditions (BC) can alter stability of modes in the Method of characteristics (MoC) employing certain ordinary-differential equation (ODE) numerical solvers. Thus, using non-periodic BC may render some of the MoC schemes stable for most practical computations, even though they are unstable for periodic BC. This fact contradicts a statement, found in some literature, that an instability detected by the von Neumann analysis for a given numerical scheme implies an instability of that scheme with arbitrary (i.e., non-periodic) BC. We explain the mechanism behind this contradiction. We also show that, and explain why, for the MoC employing some other ODE solvers, stability of the modes may be unaffected by the BC.