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Exercise 9: Derivation of FDEs for various forms of sigma and v.
Assuming that flux density are as in Exercise 5 and that and v are positive constants, use a conservation law argument to derive (a) a centered explicit FDE of the form

and (b) an ``upwind'' explicit FDE of the form

Formulate similar FDEs for the case that and v are allowed to depend on x and t; for the case that and v are allowed to depend on u.