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New example and improved examples doc page (#774)
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‎docs/examples.sh‎

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#!/bin/bash
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set -e
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MAX_HEIGHT=400
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set -e -x
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examples_md="$1/docs/documentation/examples.md"
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rm "$examples_md" || true
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casename="$(basename "$casedir")"
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if [ -f "$casedir/README.md" ]; then
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sed -e "s/\.png/-$casename-example\.png/g" "$casedir/README.md" | sed 's/^#/##/g' >> "$examples_md"
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sed -e "s/\.png/-$casename-example\.png/g" "$casedir/README.md" | sed 's/^#/##/g' | sed "s/MAX_HEIGHT/$MAX_HEIGHT/g" >> "$examples_md"
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echo '' >> "$examples_md"
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for png in $(find "$casedir" -maxdepth 1 -name '*.png'); do
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cp "$png" "$1/docs/documentation/$(basename "$png" | sed s/\.png//g)-$casename-example.png"
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done
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fi
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done
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done

‎examples/1D_inert_shocktube/README.md‎

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# 1D Multi-Component Inert Shock Tube
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References:
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Reference:
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> P. J. Martínez Ferrer, R. Buttay, G. Lehnasch, and A. Mura, “A detailed verification procedure for compressible reactive multicomponent Navier–Stokes solvers”, Comput. & Fluids, vol. 89, pp. 88–110, Jan. 2014. Accessed: Oct. 13, 2024. [Online]. Available: https://doi.org/10.1016/j.compfluid.2013.10.014
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## Initial Condition
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![Initial Condition](initial.png)
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<img src="initial.png" height="MAX_HEIGHT"/>
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## Results
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![Results](result.png)
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<img src="result.png" height="MAX_HEIGHT"/>

‎examples/1D_laxshocktube/README.md‎

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# Lax shock tube problem (1D)
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Reference: P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communications on pure and applied mathematics 7 (1) (1954) 159–193.
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Reference:
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> P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communications on pure and applied mathematics 7 (1) (1954) 159–193.
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## Initial Condition
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![Initial Condition](initial.png)
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<img src="initial.png" height="MAX_HEIGHT"/>
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## Result
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![Result](result.png)
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<img src="result.png" height="MAX_HEIGHT"/>

‎examples/1D_reactive_shocktube/README.md‎

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## Initial Condition
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![Initial Condition](initial.png)
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<img src="initial.png" height="MAX_HEIGHT"/>
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## Results
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![Results](result.png)
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<img src="result.png" height="MAX_HEIGHT"/>

‎examples/1D_shuosher_old/README.md‎

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# Shu-Osher problem (1D)
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Reference: C. W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics 77 (2) (1988) 439–471. doi:10.1016/0021-9991(88)90177-5.
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Reference:
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> C. W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics 77 (2) (1988) 439–471. doi:10.1016/0021-9991(88)90177-5.
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## Initial Condition
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![Initial Condition](initial.png)
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<img src="initial.png" height="MAX_HEIGHT"/>
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## Result
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![Result](result.png)
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<img src="result.png" height="MAX_HEIGHT"/>

‎examples/1D_titarevtorro/README.md‎

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# Titarev-Toro problem (1D)
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Reference: V. A. Titarev, E. F. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, Journal of Computational Physics 201 (1) (2004) 238–260.
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Reference:
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> V. A. Titarev, E. F. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, Journal of Computational Physics 201 (1) (2004) 238–260.
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## Initial Condition
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![Initial Condition](initial.png)
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<img src="initial.png" heiht="MAX_HEIGHT"/>
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## Result
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![Result](result.png)
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<img src="result.png" heiht="MAX_HEIGHT"/>

‎examples/2D_hardcodied_ic/README.md‎

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# 2D Hardcodied IC Example
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## Initial Condition
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## Initial Condition and Result
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![Initial Condition](initial.png)
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## Result
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![Result](result.png)
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<img src="initial.png" width="45%"/>
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<img src="result.png" width="45%"/>

‎examples/2D_ibm_cfl_dt/README.md‎

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## Result
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![Result](result.png)
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<img src="result.png" height="MAX_HEIGHT"/>

‎examples/2D_isentropicvortex/README.md‎

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# Isentropic vortex problem (2D)
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Reference: Coralic, V., & Colonius, T. (2014). Finite-volume Weno scheme for viscous compressible multicomponent flows. Journal of Computational Physics, 274, 95–121. https://doi.org/10.1016/j.jcp.2014.06.003
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Reference:
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> Coralic, V., & Colonius, T. (2014). Finite-volume Weno scheme for viscous compressible multicomponent flows. Journal of Computational Physics, 274, 95–121. https://doi.org/10.1016/j.jcp.2014.06.003
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## Density
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![Density](alpha_rho1.png)
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<img src="alpha_rho1.png" height="MAX_HEIGHT"/>
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## Density Norms
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![Density Norms](density_norms.png)
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<img src="density_norms.png" height="MAX_HEIGHT"/>

‎examples/2D_lid_driven_cavity/README.md‎

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# Lid-Driven Cavity Problem (2D)
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Reference: Bezgin, D. A., & Buhendwa A. B., & Adams N. A. (2022). JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows. arXiv:2203.13760
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> Bezgin, D. A., & Buhendwa A. B., & Adams N. A. (2022). JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows. arXiv:2203.13760
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Reference: Ghia, U., & Ghia, K. N., & Shin, C. T. (1982). High-re solutions for incompressible flow
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> Ghia, U., & Ghia, K. N., & Shin, C. T. (1982). High-re solutions for incompressible flow
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using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48, 387-411
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Video: https://youtube.com/shorts/JEP28scZrBM?feature=share
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## Final Condition
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![Final Condition](final_condition.png)
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<img src="final_condition.png" height="MAX_HEIGHT"/>
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## Centerline Velocities
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![Centerline Velocities](centerline_velocities.png)
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<img src="centerline_velocities.png" height="MAX_HEIGHT"/>

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