When is it possible to write the conservation equation $\partial_tU+ \partial_XF(U)$ (System in one dimension) in quasilinear form? What exactly is this quasilinear form and is DF(U) always linear?
Conservation laws in quasilinear form
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0If $F$ is differentiable, we can rewrite it as $\partial_t U + DF(U)\partial_x U = 0$, which is linear in $U$'s highest derivatives (here: first derivatives) and hence quasi-linear. – 2012-09-26
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0Thanks, does it mean that $DF(U)$ is always linear? – 2012-09-26
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0$DF(U)$ is, by definition of the derivative, a linear map $\mathbb R^d \to \mathbb R^d$, so $DF(U)\partial_x U$, is linear in $\partial_x U$, note that $DF(U)\partial_x U$ is in general not linear in $U$. – 2012-09-26