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## 6.3 Non-Linear equations with CIP

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The same approach is applicable more generally for non-linear and vector equations

 (1)

where and . The problem is again decomposed in alternating phases with / without advection describing the evolution of the function

 (2)

and by differentiation of (eq.6.3#eq.1), the evolution of the derivative

 (3)

Starting with the non-advection phase, the discretized function is first evolved according to

 (4)

where the super-script ( ) refers to an intermediate step. To avoid an explicit evaluation of , the equation for the derivative is computed with
 (5)

The advection phase can then be evolved in the same manner as before (eq.6.2#eq.4), by shifting the cubic-Hermite polynomials along the characteristics (exercise 6.04).

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