Second-order well-balanced Lagrange-projection schemes for blood flow equations

Abstract

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On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations

“…As it is already known, the Lagrange-projection splitting and the acoustic-transport one can be interpreted as two different ways of describing the same kind of decomposition. Indeed, it has already been illustrated the relation between the acoustic and Lagrangian numerical approximation, see for instance the following papers applied to the shallow water system [15], the gas dynamics equations [13] and the blood flow system [17]. Therefore, following the lines of these works, we present the acoustic-transport splitting for the two-layer shallow water model as it will be useful for the development of the numerical strategy.…”

“…A first difficulty is related to the fact that usually the mass variable m is introduced for the acoustic system at this stage, see for instance the previous references [15,13,17]. For example, in the shallow water system, this new variable m is defined such that ∂ m = 1 h ∂ x with h the water height.…”

“…For similar procedure applied to the shallow water system or the blood flow equations, see respectively [20] and [17]. Hence, the only difference between formulation (31) and ( 34) is related to the use of the variable X or LX in the definition of the numerical fluxes.…”

“…Concerning the numerical strategy, here we aim to design and implement well-balanced implicit-explicit Lagrange-projection schemes. So far Lagrange-Projection (LP) methods have been studied for different mathematical models as the shallow water system [15] and related models [10,11], the gas dynamic equations [13,12] and the blood flow system [17]. However, up to our knowledge, they have never been employed to numerically approximate the two-layer shallow water equations.…”

“…Last but not least, we are interested in the well-balanced property of the numerical schemes, meaning that the numerical methods are able to preserve the stationary solutions of the mathematical model, at least for the so-called lake-at-rest solutions. Indeed, it is well-known that otherwise we could observe spurious oscillations in the numerical simulations when near to a steady state, refer for instance to [8,10,11,15,17,20] and to [2] for well-balanced schemes with and without the Lagrange-projection decomposition respectively. Therefore, let us see which are the stationary solutions of the model.…”