Abstract

We present numerical schemes for solving hyperbolic balance laws, admitting stiff source terms, to high -order of accuracy in both space and time. The schemes belong to the ADER family of methods and as such rest on two building blocks, namely a non-linear spatial reconstruction procedure and the solution of a generalized Riemann problem (GRP), in which the data is piece-wise smooth and the equations include source terms. This paper presents contributions on both building blocks. Concerning spatial reconstruction we present various versions of a new Essentially Non-Oscillatory (ENO) type reconstruction, called here Averaged ENO (AENO). As to the generalized Riemann problem, we present an improved solver based on two basic ingredients, namely the implicit Taylor series expansion reported in Toro and Montecino (2015) and the simplified Cauchy-Kowalewskaya procedure reported in Montecinos and Balsar (2020).The resulting ADER schemes are implemented and systematically assessed for the linear advection equation and for non-linear hyperbolic system that governs blood flow, with a tube law admitting arteries or veins. For the linear advection equation we implement the new schemes to accuracy ranging from first to seventh order; convergence rate studies confirm that the theoretically expected accuracy is achieved for most versions of the various schemes studied. For the non-linear system we implement the new schemes to accuracy ranging from first to fifth order. Convergence rate studies based on problems with smooth solutions confirm again that the methods attain the theoretically expected accuracy. For the non-linear system the methods are also assessed for Riemann problems for blood flow in arteries with exact solution containing smooth parts and discontinuities, elastic jumps (shocks), and contact discontinuities. Furthermore, for the non-linear system the methods are also assessed for blood flow in veins for a steady problem with smooth analytical solution. As a final assessment of the potential applicability of the methods for realistic simulations in haemodynamics, we apply the methods on a network of 37 blood vessels, for which experimental data is available. The new schemes presented in this paper are simpler than existing versions of ADER methods, involving implicit Taylor series and the Cauchy-Kowalewskaya procedure, published in the literature and, overall, the computational results demonstrate that the new schemes give comparable or superior results, with respect to existing high-order schemes.