-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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