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In this paper, the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge-Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. The asymptotic accuracy is concerned with the closeness of the compressible solution with that of its incompressible counterpart in a low Mach number regime. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.

We consider two distinguished asymptotic limits of the Euler equations in a gravitational field, namely the incompressible and Boussinesq limits. Both these limits can be obtained as singular limits of the Euler equations under appropriate scaling of the Mach and Froude numbers. We propose and analyse an asymptotic preserving (AP) time discretisation for the numerical approximation of the Euler system in these asymptotic regimes. A key step in the construction of the AP scheme is a semi-implicit discretisation of the fluxes and the source term. The non-stiff convective terms are treated explicitly whereas the stiff pressure-gradient and source term are implicit. The implicit terms are combined to get a nonlinear elliptic equation. We show that the overall scheme is consistent with the respective limit system when the Mach number goes to zero. A linearised stability analysis confirms the $L^2$-stability of the proposed scheme. The results of numerical experiments validate the theoretical findings., Comment: 8 pages, 2 figures

A successful protocol was developed for mass propagation of Lawsonia inermis Linn., an important medicinal plant. Multiple shoots were induced in apical and axillary meristems derived from mature explants of L. inermis on Murashige and Skoog (1962) medium supplemented with 0.25 mg/l 6-benzylaminopurine (BA), 0.25 mg/l Kinetin (Kn), 0.5 mg/l ascorbic acid and 3% (w/v) sucrose. The rate of multiplication was higher when the cultures were incubated under continuous light rather than the 14h photoperiod. Rooting was readily achieved upon transferring the microshoots onto MS basal semi-solid medium supplemented with 0.25 mg/l indole-3-butyric acid (IBA) after ten days of culture. Micropropagated plantlets were acclimatized and successfully grown in soilSe desarrolo un protocolo exitoso para la propagación en masa de Lawsonia inermis Linn., una planta medicinal importante. Se indujeron múltiples tallos en meristamos apicales y auxiliares derivados de explantes maduros de L. inermis en medio de Murashige y Skoog (1962) suplementado con 0.25 mg/l 6-benzylaminopurina (BA), 0.25 mg/l quinetina (Kn), 0.5 mg/l ácido ascórbico y 3 % (w/v) sucrosa. La tasa de multiplicación fue más alta cuando los cultivos fueron incubados bajo luz continua que bajo el fotoperíodo de 14hr. El enraizamiento se logró al tranferir los microtallos al medio basal, semi-sólido MS suplementado con 0.25 mg/l de ácido indol-3-butiríco (IBA) luego de 10 días de cultivo. Las plántulas micropropagadas fueron aclimatizadas y cultivadas con éxito en el suelo

A successful protocol was developed for mass propagation of Lawsonia inermis Linn., an important medicinal plant. Multiple shoots were induced in apical and axillary meristems derived from mature explants of L. inermis on Murashige and Skoog (1962) medium supplemented with 0.25 mg/l 6-benzylaminopurine (BA), 0.25 mg/l Kinetin (Kn), 0.5 mg/l ascorbic acid and 3 % (w/v) sucrose. The rate of multiplication was higher when the cultures were incubated under continuous light rather than the 14 hr photoperiod. Rooting was readily achieved upon transferring the microshoots onto MS basal semi-solid medium supplemented with 0.25 mg/l indole-3-butyric acid (IBA) after ten days of culture. Micropropagated plantlets were acclimatized and successfully grown in soil. Se desarrolló un protocolo exitoso para la propagación en masa de Lawsonia inermis Linn., una planta medicinal importante. Se indujeron múltiples tallos en meristemos apicales y axilares derivados de explantes maduros de L. inermis en medio de Murashige y Skoog (1962) suplementado con 0.25 mg/l 6-benzylaminopurina (BA), 0.25 mg/l quinetina (Kn), 0.5 mg/l ácido ascórbico y 3 % (w/v) sucrosa. La tasa de multiplicación fue más alta cuando los cultivos fueron incubados bajo luz continua que bajo el fotoperíodo de 14 hr. El enraizamiento se logró al transferir los microtallos al medio basal, semi-sólido MS suplementado con 0.25 mg/l de ácido indol-3-butírico (IBA) luego de 10 días de cultivo. Las plántulas micropropagadas fueron aclimatizadas y cultivadas con éxito en el suelo.