_PPM.dox

1/*! \page PPM PPM Advection Scheme

2

3\section section_PPM Advection Scheme

4

5Following \cite colella1984 and \cite carpenter1990, we use the Piecewise Parabolic

6Method (PPM) to represent values within the model cells. Each cell is assumed to

7have a piecewise parabolic representation, which is uniquely prescribed by

8conservation and the two edge values. This method has the following features:

9

10\li The PPM approach is conservative.

11\li The (unlimited) order of accuracy is determined by the estimates of the edge

12values.

13\li Monotonicity is ensured by adjusting the edge values to flatten the profile.

14

15An example is shown in this figure:

16

17\image html ppm_arc.png The parabolic representation of a field within a cell.

18\image latex ppm_arc.png The parabolic representation of a field within a cell.

19

20\f[

21 x'_i \equiv \frac{x - x_{i-1/2}} {\Delta x_i}

22\f]

23

24\f[

25 \Delta x_i \equiv x_{i + 1/2} - x_{i- 1/2}

26\f]

27

28\f[

29 c \equiv u \Delta t / \Delta x_i

30\f]

31

32\f[

33 A_i(x') = a_L + (a_R - a_L) x'_i + a_6 x'_i(1 - x'_i)

34\f]

35

36\f[

37 a_6 = 6a_i - 3 (a_R + a_L)

38\f]

39

40\f{eqnarray}

41 a_i &= \int_0^1 A_i(x'_i) dx'_i = \int_0^1 a_L + (a_R - a_L) x'_i + a_6 x'_i (1

42 - x'_i) dx'_i \\

43 &= \left[ a_L x'_i + \frac{1}{2} (a_R - a_L) x_i^{\prime 2} + a_6 \left( \frac{1}{2}

44 x_i^{\prime 2} - \frac{1}{3} x_i^{\prime 3} \right) \right]_0^1 \\

45 &= \frac{1}{2} (a_R + a_L) + \frac{1}{6} a_6

46\f}

47

48\f{eqnarray}

49 F_{i+1/2} &= \frac{1}{\Delta t} \int_{x_{i + 1/2} - u \Delta t}^{x_{i + 1/2}}

50 A_i^n(x) dx =

51 \frac{\Delta x}{\Delta t} \int_{1-c}^1 A_i (x'_i) dx'_i \\

52 &= \frac{\Delta x}{\Delta t} \left[ a_L x'_i + \frac{1}{2} (a_R - a_L)

53 x_i^{\prime 2} + a_6 \left( \frac{1}{2} x_i^{\prime 2} -

54 \frac{1}{3} x_i^{\prime 3} \right) \right]_{1 - c}^1 \\

55 &= \frac{\Delta x}{\Delta t} \left[ a_L c + (a_R - a_L + a_6) \left( c -

56 \frac{1}{2} c^2 \right) - a_6 \left( c - c^2 + \frac{1}{3} c^3 \right) \right] \\

57 &= u \left[ a_R + \frac{1}{2} (a_L - a_R) c + a_6 \left( \frac{1}{2} c -

58 \frac{1}{3} c^2 \right) \right]

59\f}

60

61The choice of \f$a_L\f$ and \f$a_R\f$ is not unique, but can be done according to

62\cite colella1984 (CW84) or \cite huynh1997 (H3) as mentioned in \ref

63Tracer_Advection.

64

65*/