By François Bouchut
This e-book is dedicated to finite quantity equipment for hyperbolic platforms of conservation legislation. The accessory is wear the improvement of instruments for interpreting the nonlinear balance of Godunov schemes. ranging from theoretical concerns, the schemes are derived until eventually a really sensible point, assembly a few required positive aspects resembling for instance the therapy of vacuum in gasoline dynamics. with regards to resources, the final notions of consistency, order of accuracy and well-balancing are built, and utilized to the development of powerful schemes.
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Extra info for Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources
8. The half CFL condition can be replaced by a usual CFL condition if one uses the notion of maximum principle by interface (observe that the maximum principle is a form of invariant domain property). 9. Another passive transport scheme can also be considered in the context of ﬂux vector splitting schemes. 217) comes from a 0+ 0− 0 = Fi+1/2 + Fi+1/2 . 191), then we have a decomposition Fi+1/2 The condition for the scheme to preserve nonnegativity of ρ implies then that 0+ 0− ≥ 0, Fi+1/2 ≤ 0. The natural scheme in the FVS spirit is then given by Fi+1/2 φ 0+ 0− Fi+1/2 = Fi+1/2 φi + Fi+1/2 φi+1 .
167), ρ, e, s, π are understood as independent variables. 158), the equation on ρs is taken as conservative variable and the energy e is just forgotten at time tn+1 , being replaced by e(ρn+1 , sn+1 ). On the contrary, in the case of true gas dynamics, e is kept while s is replaced by s(ρn+1 , en+1 ). The last variable π is in any case replaced by the true pressure, p(ρn+1 , sn+1 ) or p(ρn+1 , en+1 ). 141), to which we prescribe s∗l = sl , s∗r = sr . Only the initialization is modiﬁed, now πl = pl = p(ρl , sl ), πr = pr = p(ρr , sr ).
7), we deduce the following result. 21. 11). It has the following properties: (i) it preserves the nonnegativity of ρ, (ii) it preserves the positivity of e, (iii) it satisﬁes discrete entropy inequalities, (iv) if satisﬁes the maximum principle on the speciﬁc entropy. (v) stationary contact discontinuities where u = 0, p = cst are exactly resolved. (vi) it handles data with vacuum. 140). Only the property (v) has not been discussed previously. 167) with π = p. In this property, the choice of cl and cr does not matter at all.