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

Abstract

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(Δt3+h4) convergence in ℓ∞(0,T;ℓ2) norm, which leads to the necessary ℓ∞ error estimate using the inverse inequality, under a standard constraint Δt≤Ch. A numerical accuracy check is given and some numerical simulation results are also presented.