A fourth-order compact functional-calculus method for distributed-order spectral diffusion

We studied a full-spectral distributed-order diffusion equation on a rectangular domain with homogeneous Dirichlet boundary conditions, driven by a nonnegative integrable order-density function that represents a continuum of spatial scaling contributions. The order superposition was represented by the equation $\varphi(s) = \int_{\mu^-}^{\mu^+}\rho(\mu)s^{\mu/2}\, \, \mathrm{d}\mu$ and our objective was to construct and analyze a fourth-order discretization that preserves this spectral structure while admitting an efficient sine-modal implementation. We discretized the positive Dirichlet Laplacian by the fourth-order compact Kronecker-sum operator $ A^{(4)}_{\Delta, h} = I_y\otimes A^{(4)}_{x, h}+ A^{(4)}_{y, h}\otimes I_x$, defined $ \Phi_h = \varphi(A^{(4)}_{\Delta, h})$, and used the dissipative generator $ L_h = - \Phi_h$. The main consistency result showed that the compact eigenvalue perturbation is preserved under the distributed-order functional calculus, yielding a fourth-order discrete $L^2$ approximation to $\varphi(-\Delta_D)$ after grid restriction. An unsplit modal Crank–Nicolson scheme was then obtained by diagonalizing $ \Phi_h$ with the discrete sine transform, where the positivity of the exact symbol or the positive quadrature symbol gives unconditional discrete $L^2$ stability. Under the stated regularity assumptions, the fully discrete error is $O(\tau^2+h^4)$ for the exact-symbol scheme and $O(\tau^2+h^4+E_\mu)$ for its positive-quadrature version. The numerical experiments were consistent with the proved fourth-order spatial accuracy, second-order temporal accuracy, and unconditional stability, and they also illustrated quadrature sensitivity, implementation scaling, and the accuracy gain over the second-order full-spectral counterpart.