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Update _docs_v7/Slope-Limiters-and-Shock-Resolution.md
Co-authored-by: Pedro Gomes <[email protected]>
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_docs_v7/Slope-Limiters-and-Shock-Resolution.md

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## Theory: An introduction to slope limiters
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For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative.
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For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities is a common occurrence. The use of high-order numerical schemes is desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flow field. However, high-resolution linear schemes often result in numerical oscillations near the shock due to the high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that significantly degrade the accuracy of your solution and pollute the domain. An example of this phenomenon is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative.
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<img src="../../docs_files/LW_example.png" width="500">
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