High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices
The author considers linear hyperbolic systems in \(d\geq 1\) space dimensions along with initial conditions and, on inflow boundaries, first order data in a form remembering the Roe averages. The aim is to avoid cumbersome inversion of global mass matrices when working with globally continuous finite element approximations in space. For this aim, a procedure of time stepping is presented which approximates the Picard iteration using a first-order operator (element-wise Euler forward) and gaining higher order with the help of a finite number of steps of deferred correction iteration defined using a higher-order explicit time integration method. The author shows the Lipschitz continuity of the spatial residuals appearing in the first-order operator and proves also the invertibility of the latter. He further clarifies conditions on the choice of basis functions for the spatial approximation (which exclude Lagrange elements). In his numerical experiments, results for one- and two-dimensional examples are reported which include also discontinuous solutions and a nonlinear case. Even though the paper is said to constitute an extended version of two earlier papers, the presentation contains rather superficial details both in the use of language (e.g., ``the last one for the one dimensional for the one dimensional case'', p. 479) and mathematically (e.g., in formula (9) on p. 465, the last component of the operator \(\mathcal L^1\) is defined as \(y_1-y_0-\Delta t \int_{t_{n,0}}^{t_{n,1}}{\mathcal I}_0(f;s)ds\) which means \(y_1-y_0-\xi_1\Delta t^2 f(y_0,t_{n,0})\) and also contradicts formula (8) which instead of \(\xi_1\Delta t^2 f(y_0,t_{n,0})\) exhibits \(\xi_0\Delta t f(y_0,t_{n,0})\) -- but both versions are wrong).
scientific article; zbMATH DE number 6826581
scientific article