Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving hyperbolic systems. Part I: Periodic boundary conditions

Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving hyperbolic systems. Part II: Nonreflecting boundary conditions

“…In [1] we showed that for periodic BC, the MoC with any of those three ODE solvers exhibits numerical instability. For the SE solver, the strongest such an instability occurs in the "ODE" and "anti-ODE" limits, i.e.…”

“…The component form and a non-constant, soliton/kink solution of (3) can be found in, e.g., [1]. Here (as in [1]) we consider the numerical stability of a constant solution of (3):…”

“…when it is simulated by the MoC. Considering only one representative, (3), of a broader class of models (see [1]) and its simplest solution, (4), will allow us to focus on the analysis of the numerical scheme without being distracted by complexities of the physical model. The relevance of the numerical (in)stability of the constant solution (4) to the numerical (in)stability of "more physically interesting", soliton/kink and related, solutions was discussed in [1] after Eq.…”

“…In Part I [1] of this study we considered the numerical stability of the Method of characteristics (MoC) applied to a class of hyperbolic partial differential equations (PDEs) with periodic boundary conditions (BC). More specifically, the ordinary differential equation (ODE) numerical solvers employed by the MoC in [1] along the characteristics were the simple Euler (SE), modified Euler (ME), and Leapfrog (LF) ones. The class of the PDEs considered in [1] has the linearized form:…”

“…More specifically, the ordinary differential equation (ODE) numerical solvers employed by the MoC in [1] along the characteristics were the simple Euler (SE), modified Euler (ME), and Leapfrog (LF) ones. The class of the PDEs considered in [1] has the linearized form:…”