An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term.

We present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term. The solution to such an equation is proven to have a conservative nonlinear energy. Due to the special nature of the nonlinear term, the positivity is proven to be preserved under a periodic boundary condition for the solution. For the numerical scheme, a highly nonlinear fractional term is involved, for the theoretical justification of the energy stability. We propose a linear iteration algorithm to solve this nonlinear numerical scheme. A theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step. We also provide a detailed convergence analysis for the second order scheme, in the ℓ ∞ ( 0 , T ; ℓ ∞ ) norm. Such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution. As a result, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O ( Δ t 3 + h 4 ) convergence in ℓ ∞ ( 0 , T ; ℓ 2 ) norm, which leads to the necessary ℓ ∞ error estimate using the inverse inequality, under a standard constraint Δ t ≤ C h . A numerical accuracy check is given and some numerical simulation results are also presented. [ABSTRACT FROM AUTHOR]