Stable and flux-conserved meshfree formulation to model shocks

Abstract

Accurate shock modeling requires that two critical issues be addressed: (1) correct representation of the essential shock physics, and (2) control of Gibbs phenomenon oscillation at the discontinuity. In this work a stable (oscillation limiting) and flux-conserved formulation under the reproducing kernel particle method is developed for shock modeling. A smoothed flux divergence is constructed under the framework of stabilized conforming nodal integration, which is locally-enriched with a Riemann solution to satisfy the entropy production constraints. This Riemann-enriched flux divergence is embedded into the reproducing kernel formulation through a velocity correction that also provides oscillation control at the shock. The correction is constrained to the shock region by an automatic shock detection algorithm that is constructed using the intrinsic spectral decomposition feature of the reproducing kernel approximation. Several numerical examples are provided to verify accuracy of the proposed formulation.

Appendix 1

The jump condition is satisfied by the weak formulation of the governing equation in conservative form, which is shown by considering equation (23) with integration over the arbitrary space-time domain. The one dimensional space-time domain $\Upsilon$, as shown in Fig. 6 is considered for simplicity.

$$\begin{aligned} \mathop \int \nolimits _{t=0}^\infty \mathop \int \nolimits _{x=-\infty }^\infty w\left( {x,t} \right) \left[ {u\left( {x,t} \right) ,_t +f\left( {u\left( {x,t} \right) } \right) ,_x } \right] dx\,dt=0\nonumber \\ \end{aligned}$$

(62)

The test function is bounded according to the $H_0^1$ space, and it is referred to as a compact test function with the properties $w\left( {\infty ,t} \right) =w\left( {-\infty ,t} \right) =w\left( {x,\infty } \right) =0$. The initial condition is assumed to be $w\left( {x,0} \right) =0$. By applying integration by parts to $\mathop \int \nolimits _0^\infty w\left( {x,t} \right) u\left( {x,t} \right) ,_t dt$ and $\mathop \int \nolimits _{-\infty } ^\infty w\left( {x,t} \right) f\left( {u\left( {x,t} \right) } \right) ,_x dx$ in Eq. (62), the following is obtained

$$\begin{aligned}&\mathop \int \nolimits _0^\infty w\left( {x,t} \right) u\left( {x,t} \right) ,_t dt\nonumber \\&\quad =\left. {w\left( {x,t} \right) u\left( {x,t} \right) } \right| _{t=0}^\infty -\mathop \int \nolimits _0^\infty w\left( {x,t} \right) ,_t u\left( {x,t} \right) dt\end{aligned}$$

(63)

$$\begin{aligned}&\quad =-\mathop \int \nolimits _0^\infty w\left( {x,t} \right) ,_t u\left( {x,t} \right) dt\end{aligned}$$

(64)

$$\begin{aligned}&\mathop \int \nolimits _{-\infty }^\infty w\left( {x,t} \right) f\left( {u\left( {x,t} \right) } \right) ,_x dx\nonumber \\&\quad =\left. {w\left( {x,t} \right) f\left( {u\left( {x,t} \right) } \right) } \right| _{x=-\infty }^\infty \nonumber \\&\qquad -\mathop \int \nolimits _{-\infty }^\infty w\left( {x,t} \right) ,_x f\left( {u\left( {x,t} \right) } \right) dx\nonumber \\&\quad =-\mathop \int \nolimits _{-\infty }^\infty w\left( {x,t} \right) ,_x f\left( {u\left( {x,t} \right) } \right) dx \end{aligned}$$

(65)

Consequently (62) becomes

$$\begin{aligned}&\mathop \int \nolimits _{t=0}^\infty \mathop \int \nolimits _{x=-\infty }^\infty w\left( {x,t} \right) ,_t u\left( {x,t} \right) +w\left( {x,t} \right) ,_x\nonumber \\&\quad \times f\left( {u\left( {x,t} \right) } \right) dx\,dt=0 \end{aligned}$$

(66)

In the presence of a shock, a discontinuity forms in the $x-t$ domain along the boundary $\varGamma _s$ such that the space-time domain is divided into two smooth sub-domains, $\Upsilon ^{-}$ and $\Upsilon ^{+}$ as shown in Fig. 6. Normal to the discontinuity is $\mathbf{\mathfrak {n}}=\left[ {n_x ,n_t } \right]$, which is a function of space and time. Solutions to the left and right of the discontinuity are denoted as $u^{-}\left( {x,t} \right)$ and $u^{+}\left( {x,t} \right)$, respectively. Considering the presence of a discontinuity, Eq. (66) can be written as

$$\begin{aligned}&\mathop \int \nolimits _{\Upsilon ^{-}} w\left( {x,t} \right) ,_t u\left( {x,t} \right) +w\left( {x,t} \right) ,_x f\left( {u\left( {x,t} \right) } \right) dx\,dt\nonumber \\&\quad +\,\mathop \int \nolimits _{\Upsilon ^{+}} w\left( {x,t} \right) ,_t u\left( {x,t} \right) \nonumber \\&\quad +\,w\left( {x,t} \right) ,_x f\left( {u\left( {x,t} \right) } \right) dx\,dt=0 \end{aligned}$$

(67)

Again using integration by parts and the divergence theorem and making use of the weak form expression of Eq. (23) applied to the smooth sub-domains, Eq. (67) is transformed to the contour integral over the discontinuity

$$\begin{aligned}&\mathop \int \nolimits _{\Gamma _s } w\left( {x,t} \right) \left[ {u^{-}\left( {x,t} \right) n_t^- +f\left( {u^{-}\left( {x,t} \right) } \right) n_x^- } \right] d\Gamma \nonumber \\&\quad =-\mathop \int \nolimits _{\Gamma _s } w\left( {x,t} \right) \left[ {u^{+}\left( {x,t} \right) n_t^+ +f\left( {u^{+}\left( {x,t} \right) } \right) n_x^+ } \right] d\Gamma \nonumber \\ \end{aligned}$$

(68)

Due to arbitrariness of the test function and considering that ${\mathbf{\mathfrak {n}}}^{-}=-{\mathbf{\mathfrak {n}}}^{+}$, Eq. (68) implies

$$\begin{aligned}&u^{-}\left( {x,t} \right) n_t +f\left( {u^{-}\left( {x,t} \right) } \right) n_x\nonumber \\&\quad =u^{+}\left( {x,t} \right) n_t +f\left( {u^{+}\left( {x,t} \right) } \right) n_x \end{aligned}$$

(69)

The shock velocity, $\xi$, is related to ${\mathbf{\mathfrak {n}}}$ by $\xi =dx/dt=-n_t /n_x$, which is used with Eq. (69) to obtain

$$\begin{aligned} \xi =\frac{f\left( {u^{-}\left( {x,t} \right) } \right) -f\left( {u^{+}\left( {x,t} \right) } \right) }{u^{-}\left( {x,t} \right) -u^{+}\left( {x,t} \right) } \end{aligned}$$

(70)

The Rankine–Hugoniot jump equation is given by Eq. (33), and shows that the jump condition is naturally obtained from the weak form of the conservation equation in (70).