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MA557/MA578/CS557 Lecture 6

MA557/MA578/CS557 Lecture 6. Spring 2003 Prof. Tim Warburton timwar@math.unm.edu. Next Lecture (6). Alternative flux formulations Alternative time stepping schemes. In detail:. The time rate of change of total mass in the section of pipe [a,b].

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MA557/MA578/CS557 Lecture 6

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  1. MA557/MA578/CS557Lecture 6 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

  2. Next Lecture (6) • Alternative flux formulations • Alternative time stepping schemes

  3. In detail: The time rate ofchange of total mass in the section of pipe [a,b] The flux out of the section at the right end of the section of pipe per unit time The flux into the section at the left end of the section of pipeper unit time

  4. Recall We Wish to Discretize: • We will consider alternatives for discretizing the time derivative on the left hand side • We will consider alternatives for discretizing the pointvalues of the density on the right hand side.

  5. Lax-Friedrichs Fluxes • Consider the alternative: Note left biased jump term replaced with centered jump term

  6. Numerical Scheme (LF fluxes)

  7. Numerical Scheme Stability (LF fluxes)

  8. Summary of LF Stability • This basically shows that the LF fluxes formulation leads to a contraction map – although now we have the slightly trickier issue of dealing with the outflow boundary at xN. • To get around this we would most likely drop to the upwind scheme at xN. • In that case the scheme satisfies the contraction condition as before (assuming the error for the 0-cell is zero) • CFL stability condition is

  9. Consistency For LF

  10. Local Truncation Error • Suppose at the beginning of a time step we actually have the exact solution -- one question we can ask is how large is the error committed in the evaluation of the approximate solution at the end of the time step. • i.e. choose • Then

  11. Taylor Series • We expand about xi,tn with Taylor series:

  12. Estimating Truncation Error • Inserting the formulas for the expanded q’s:

  13. Estimating Truncation Error • Removing canceling terms:

  14. Simplifying

  15. Rearranging

  16. Local Truncation Error: LF Leading order behavior of local truncation error is again O(dx)

  17. Consistency + Stability • By showing that the LF formulation is consistent (R->0 as dt->0) and contraction mapping we have proved convergence…

  18. Class Group Exercise • Prove the following is consistent and stable: • Lax-Wendroff formula: • Describe any difficulties with boundary terms… • Estimate the order of accuracy • Volunteer to write results on the board 

  19. Lecture 7 • Introducing the Discontinuous Galerkin Method for the advection equation. • Review of some special univariate polynomials.

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