H3N3-2\(_\sigma \)-based difference schemes for time multi-term fractional diffusion-wave equation

In this work, we design an H3N3-2 \(_\sigma \) $_{\sigma }^{}$ approximation formula for the linear combination of the multi-term Caputo derivatives \(\sum \nolimits _{r=0}^m\,\lambda _r{\,}_C\textrm{D}_{0,t}^{\alpha _r} p(t),\) ${\sum }_{r=0}^m{\lambda }_r_C^{}{\text{D}}_{0,t}^{{\alpha }_r}p(t),$ where \(m\ge 1,\) $m\ge 1,$ and \(\lambda _r\) ${\lambda }_r$ ’s are positive constants, \(1<\alpha _m<\alpha _{m-1}<\cdots<\alpha _0<2.\) $1<{\alpha }_m<{\alpha }_{m-1}<\cdots <{\alpha }_0<2.$ We determine a super-convergence point \(\sigma \) $\sigma$ using Newton iteration method, which makes the derived numerical formula can achieve at least the second-order accuracy. We also show the coefficients’ properties of the H3N3-2 \(_\sigma \) $_{\sigma }^{}$ formula. This formula is then used to numerically solve a time multi-term fractional diffusion-wave equation in multiple dimensions. The stability and convergence of the derived difference scheme in the sense of \(L^2\) $L^2$ -norm are analyzed. As a supplement, we also construct a numerical difference scheme on the graded meshes for the time multi-term fractional diffusion-wave model whose solution has an initial weak regularity. Numerical examples are provided to verify the effectiveness of the proposed difference schemes.