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On a nonlinear energy-conserving scalar auxiliary variable (SAV) model for Riesz space-fractional hyperbolic equations. (English) Zbl 1475.65071

Summary: In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
35L05 Wave equation
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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