0
$\begingroup$

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?

  • 0
    If $F$ 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
  • 0
    Thanks, does it mean that $DF(U)$ is always linear?2012-09-26
  • 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

1 Answers 1