This repository contains the implementation and documentation for simulating the Sod Shock Tube and the Sedov Blast Wave problems using Smoothed Particle Hydrodynamics (SPH). These benchmarks are fundamental in computational fluid dynamics and demonstrate the accuracy and robustness of SPH methods in shock-capturing and energy dissipation.
- Density Computation: Iterative calculation of density using adaptive smoothing lengths (
h) with a Newton-Raphson scheme. - Momentum and Energy Conservation: Includes dissipative terms for shock handling with velocity-dependent signal propagation.
- Higher-Order Kernels: Support for compact kernels like the M5 quartic spline to improve resolution and reduce bias.
- Newton-Raphson Iteration: For consistent coupling between smoothing length (
h) and density (ρ). - Shock Capturing: Dissipative terms in momentum and energy to handle shocks efficiently.
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Density Computation: [ \rho_a = \sum_b m_b W(\mathbf{r}_a - \mathbf{r}_b, h_a) ]
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Momentum Conservation: [ \frac{d\mathbf{v}a}{dt} = -\sum_b m_b \left[ \frac{P_a}{\Omega_a \rho_a^2} \nabla_a W{ab}(h_a)
- \frac{P_b}{\Omega_b \rho_b^2} \nabla_a W_{ab}(h_b) \right] ]
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Energy Conservation: [ \frac{du_a}{dt} = \frac{P_a}{\Omega_a \rho_a^2} \sum_b m_b (\mathbf{v}_a - \mathbf{v}b) \cdot \nabla_a W{ab}(h_a) ]
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Dissipative Terms (Shock Capturing): [ \Pi_\text{shock}^a = -\sum_b m_b \left[ \frac{q_{ab}^a}{\rho_a^2 \Omega_a} \nabla_a W_{ab}(h_a)
- \frac{q_{ab}^b}{\rho_b^2 \Omega_b} \nabla_a W_{ab}(h_b) \right] ]
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Sod Shock Tube:
- Solves a one-dimensional Riemann problem with discontinuities in density and pressure.
- Validates the ability of the SPH method to capture shocks and contact discontinuities.
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Sedov Blast Wave:
- Models a spherical shock wave expanding from a central explosion.
- Tests the robustness of SPH in extreme conditions and highly dynamic systems.
The repository supports the following kernel formulations:
- Cubic Spline Kernel
- M5 Quartic Kernel for increased accuracy and reduced bias.
- Clone the repository:
git clone https://github.com/your-username/your-repository.git