fix Readme.md

README.md

@@ -19,9 +19,11 @@ Currently, the development is in its initial stages, with the following function
 In kinetic theory, methods simulate the macroscopic motion of fluids by describing the evolution of distribution functions of fluid particles in velocity space over time. Such methods include direct simulation Monte Carlo (DSMC), discrete velocity method (DVM), lattice Boltzmann equation (LBE), gas-kinetic scheme, semi-Lagrangian method, implicit-explicit (IMEX) method, and others.
 ### Distribution Function
 In kinetic methods, the state of the fluid is described by a distribution function $f(\mathbf{x},\mathbf{u},t)$. The distribution function represents the number of particles at a given moment in time at a specific physical space point in a particular velocity space element. Its normalization property is
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 $$
 \rho(\mathbf{x},t)=\iiint_{-\infty}^{\infty}f(\mathbf{x},\mathbf{u},t)d\mathbf{u}
 $$, where $\rho(\mathbf{x},t)$ is the density of flows at $\mathbf{x}$ in time $t$. At higher temperatures, molecular internal degrees of freedom are excited, and the distribution function will depend on these degrees of freedom as well. For simplicity, we do not currently consider this situation.
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 <div style="text-align:center">
     <img src="https://i.postimg.cc/pTsjgW3T/20240703151108.jpg)](https://postimg.cc/ygSWRH14" alt="Image 1" width="300" title="distribution function of 2-D" style="margin-right: 1px;">
 </div>
@@ -38,24 +40,34 @@ $$
 
 ### Boltzmann Equation
 The evolution of the distribution function follows the Boltzmann equation.
-$$\frac{\partial f}{\partial t}+\mathbf{u}\cdot\frac{\partial f}{\partial \mathbf{x}}+\mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{u}}=\iiint_{-\infty}^{\infty}\int_0^{4\pi}(f^*f_1^*-ff_1)c_r\sigma d\Omega dc_1$$
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+$$
+\frac{\partial f}{\partial t}+\mathbf{u}\cdot\frac{\partial f}{\partial \mathbf{x}}+\mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{u}}=\iiint_{-\infty}^{\infty}\int_0^{4\pi}(f^*f_1^*-ff_1)c_r\sigma d\Omega dc_1
+$$
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 <!-- The Boltzmann equation holds a central position in kinetic theory. In the free molecular flow regime, the collisionless Boltzmann equation is applied, or its equilibrium solution, the Maxwellian distribution, is directly used. In the slip flow regime, the Boltzmann equation is expanded via the Chapman-Enskog method to obtain the first-order approximation, the Navier-Stokes equations, or the second-order approximation, the Burnett equations. Alternatively, a systematic asymptotic expansion yields a set of fluid mechanics equations. In the transition regime, solving flow problems involves approaches derived from the Boltzmann equation or its equivalent methods. -->
 The Boltzmann equation holds a central position in kinetic theory, which is an integro-differential equation, with a four-fold integral term on the right-hand side. The complexity introduced by this term makes solving the Boltzmann equation challenging. Therefore, as a simplification, kinetic methods often solve its model equations instead.
 ### Shakhov Model Equation
 The simplified equation obtained after simplifying the collision term is known as the Shakhov model equation:
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 $$
 \begin{cases}
 \frac{\partial f}{\partial t}+\mathbf{u}\cdot\frac{\partial f}{\partial \mathbf{x}}+\mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{u}}=\frac{g^+-f}{\tau}\\
 g^+=g[1+\frac45(1-\mathrm{Pr})\lambda^2\frac{\mathbf{u}\cdot\mathbf{q}}{\rho}(2\lambda \mathbf{u}^2-5)]
 \end{cases}
-$$, Where $\mathrm {Pr}$ is the Prandtl number, $\mathbf{q}$ is the heat flux, and $\tau$ is the relaxation time typically given through phenomenological models to match experimental data. The Shakhov model can provide results close to the Boltzmann equation for mass, momentum, energy, and heat fluxes in situations where flow velocities are not very high and deviations from equilibrium are not very large. And the current solver is based on Shakhov model.
+$$
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+, Where $\mathrm {Pr}$ is the Prandtl number, $\mathbf{q}$ is the heat flux, and $\tau$ is the relaxation time typically given through phenomenological models to match experimental data. The Shakhov model can provide results close to the Boltzmann equation for mass, momentum, energy, and heat fluxes in situations where flow velocities are not very high and deviations from equilibrium are not very large. And the current solver is based on Shakhov model.
 
 ## Computing Methods
 ### Finite Volume Method (FVM)
 The underlying discretization of the Boltzmann equation is based on finite volume method, with its discrete form as follows:
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 $$
 [\bar{U}_j\Omega_j]^{n+1}=[\bar{U}_j\Omega_j]^n-\Delta t\sum_{\partial\Omega}\mathbf F^{*}\cdot\Delta\mathbf{S}+\Delta t \bar{Q}_j\Omega_j
-$$, where $\bar U_j^n\coloneqq\frac{1}{\Omega_j}\int_{\Omega_j}Ud\Omega_j|^n$, $\bar Q_j^n\coloneqq\frac{1}{\Omega_j}\int_{\Omega_j}Qd\Omega_j$ are the cell-averaged conservative variable and source. And $\bar Q_j^*$ and $\mathbf{F}^*$ are respectively cell- and time-averaged sources and numerical flux, which are defined as $\bar Q_j^*\coloneqq\frac{1}{\Delta t}\int_n^{n+1}\bar Q_jdt$ and $\mathbf F^*\cdot \Delta \mathbf{S}\coloneqq \frac{1}{\Delta t}\int_n^{n+1}\mathbf F\cdot\Delta \mathbf{S}dt$.
+$$
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+, where $\bar U_j^n\coloneqq\frac{1}{\Omega_j}\int_{\Omega_j}Ud\Omega_j|^n$, $\bar Q_j^n\coloneqq\frac{1}{\Omega_j}\int_{\Omega_j}Qd\Omega_j$ are the cell-averaged conservative variable and source. And $\bar Q_j^*$ and $\mathbf{F}^*$ are respectively cell- and time-averaged sources and numerical flux, which are defined as $\bar Q_j^*\coloneqq\frac{1}{\Delta t}\int_n^{n+1}\bar Q_jdt$ and $\mathbf F^*\cdot \Delta \mathbf{S}\coloneqq \frac{1}{\Delta t}\int_n^{n+1}\mathbf F\cdot\Delta \mathbf{S}dt$.
 It is an exact relation for the time evolution of the space averaged conservative avriables over cell $j$ from time step $n$ to $n+1$. And the numerical flux $\mathbf F^*$ identifies completely a scheme by the way it approximates the time-averaged physical flux along each cell face.
 ### Numerical Flux
 The numerical scheme that the solver currently supports is UGKS, with interface variables obtained from VanLeer reconstruction. For specific implementation details, please refer to the relevant literature.
@@ -65,6 +77,7 @@ DVM is one of the popular approaches for solving rarefied flow problems. In this
 
 ### Adaptive Mesh Refinement (AMR)
 Considering the above limitations of DVM, we adopt Adaptive Mesh Refinement (AMR) to improve solving efficiency. AMR reallocates computational resources based on flow features, balancing efficiency and accuracy. 
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 <div style="text-align:center">
     <img src="https://i.postimg.cc/FzQx9cSD/PV-AMR.png" alt="Image 1" title="Simultaneous AMR" width="700" style="margin-right: 1px;">
 </div>