Van Leer’s methods for high-resolution differencing methods are based fundamentally on polynomial interpolation for accuracy and then monotonicity preservation (i.e., nonlinear stability). This section will show the basic forms used in Van Leer’s 1977 paper and several extensions we will present as extensions. Each of these will interpolate an analytical function over a set of five cells and we will compare the integrated error over these five cells as a measure of the accuracy of each. The function we use has both sharp unresolved and smoother oscillatory characteristics, so should show the basic nature of the interpolations. The function is 2 + \frac{1}{2}\sin(\frac{1}{4}(x-\frac{1}{2}))+\tanh(3(x+\frac{1}{4})).

Function for displaying polynomial reconstructions

Function for displaying polynomial reconstructions

The first reconstruction we show is the piecewise constant, p(\theta) = P_0 = u_j, which is the foundation of Godunov’s method, a first-order, but monotone method. The integrated difference between the analytical function and this piecewise polynomial is 0.882015, which will serve as a nice measure of success with higher order reconstructions.

Piecewise Constant – Basis of Godunov's MethodPiecewise Constant – Basis of Godunov’s Method

The next reconstruction is the piecewise linear scheme based on a centered slope computed from cell-centered values. The polynomial is P(\theta) = P_0 + P_1 \theta= u_j + s_j \theta . The improvement over the piecewise constant is obvious and the integrated error is now, 0.448331 about half the size of the piecewise constant.

Piecewise Linear – Van Leer's Scheme 1

Piecewise Linear – Van Leer’s Scheme 1 and Scheme 2

The piecewise linear based on moments is even better, cutting the error in half again to 0.210476.

Piecewise Linear based on first moments – Van Leer's Scheme 3

Piecewise Linear based on first moments – Van Leer’s Scheme 3

The next set of plots will be based on parabolas, P(\theta) = P_0 + P_1\theta +P_2(\theta^2 -\frac{1}{12}). We stay with Van Leer’s use of Legendre polynomials because they are mean preserving without difficulty. The first is the parabola determined by the three centered cell center values, which gives a relatively large error of 0.427028 almost as large as the piecewise linear interpolation.

Piecewise Parabolic – Van Leer's Scheme 4

Piecewise Parabolic – Van Leer’s Scheme 4

Our second parabola is much better, it is based on a single cell-centered value and the edge values. It gives an error of 0.08038, which is much better than the last parabola (by a factor of five!).

Piecewise Parabolic average plus edges – Van Leer's Scheme 5

Piecewise Parabolic average plus edges – Van Leer’s Scheme 5

Using the second moment provides even better results and lowers the error to half that of the center-edge reconstruction, 0.04i887. As we will see this result cannot be improved upon too greatly.

Piecewise Parabolic – based on two moments, Van Leer's Scheme 6

Piecewise Parabolic – based on two moments, Van Leer’s Scheme 6

Now we increase the order of reconstruction to piecewise cubic, P(\theta) = P_0 + P_1 \theta + P_2 (\theta^2 - \frac{1}{12}) + P_3 \theta^3. We will look at two reconstructions, the first based on three cell-centers, and an integral of the derivative in the center cell. This is a Hermite scheme, and based on previous experience with Schemes 1 and 4 we should expect its performance to be relatively poor with an error of 0.14884, one-third of that found with the piecewise parabolic scheme 4. The second cubic reconstruction will use the cell-center, edges and the first moment, provides excellent results, 0.044014 approximately on par with the two moment piecewise parabola the basis of scheme 6. The method to update the degrees of freedom is arguably simpler (and the overall degrees of freedom is equivalent).

Piecewise Cubic – Using slopes our first new scheme

Piecewise Cubic – Using slopes our first new scheme

Piecewise Cubic – using moments our second scheme

Piecewise Cubic – using moments our second scheme

I will now finish the post with the last set of reconstructions based on piecewise quartics, P(\theta) = P_0 + P_1 \theta + P_2 (\theta^2 - \frac{1}{12}) + P_3 \theta^3 + P_4 (\theta^4 -\frac{1}{80}). Four different methods will be used to determine the coefficients. The first is a fairly important approach because the evaluation provides the stencil used for the upstream-centered fifth order WENO flux. This polynomial is determined by the cell-centered values in the cell and the two cells to the left or right. The figure shows that the polynomial is relatively poor in terms of accuracy, and confirmed by the integrated error, 0.404114 barely better than the parabola used for scheme 4. The quartic reconstruction must provide greater value.

Piecewise Quartic – Cell Centers only basis of the classical fifth-order WENO edge scheme

Piecewise Quartic – Cell Centers only basis of the classical fifth-order WENO edge scheme

The second is based on three cell centers and two edge values, and is an obvious extension of Scheme 5. This method delivers significantly better results with an integrated error of 0.090751. This is actually disappointing when compared with the parabola for schemes 5 and 6. Clearly the roughness of the function is a problem. This strongly points at the value of compact schemes, and nonlinear stability mechanism.

Piecewise Quartic – three cell centers and edge values

Piecewise Quartic – three cell centers and edge values

We close with a scheme based on cell center, with edge derivatives and slopes. This method is quite compact and we expect good results. It provides the best results thus far with an integrated error of 0.0145172 about a third of the scheme 6 reconstruction with the same number of degrees of freedom. I will note that Phil Roe suggested this approach to me noting that it was outstanding. He is right.

Piecewise Quartic –one cell-center, edges and edge derivatives

Piecewise Quartic –one cell-center, edges and edge derivatives

We look at one last scheme with a quartic reconstruction using the first and second moments, the cell-center value, and the edges. The extra degree of freedom produces outstanding results with an integrated error of 0.00726529 half the size of the error with the previous scheme.

A piecewise quartic reconstruction based on the cell-center, first and second cell-centered moments and edge values. This is the best reconstruction shown here.

A piecewise quartic reconstruction based on the cell-center, first and second cell-centered moments and edge values. This is the best reconstruction shown here.