Given the broader spectrum of variables to choose from new schemes with exceptional properties can be developed.  One can blend volume-averaged, point, moment and derivatives together.  These schemes are all compact and have exceptional numerical properties.  We introduced them earlier as the extensions of the basic interpolations employed by Van Leer, but using the same principles.

On thing to recognize immediately is that the famous PPM scheme is a cell-centered implementation of the basic approach associated with Scheme V.  The difference is the construction of edge-centered data from cell-centered quantities.  The way that the original PPM overcame the issue was to use higher order approximations for the cell-edge quantities, specifically fourth-order edges for what is an intrinsically third-order scheme.  This allows the approximation to asymptotically approach fourth-order accuracy in the limit of vanishing Courant number.

More recently fifth, sixth and seventh order approximations have been used with PPM to good effect.  These have been applied to the use of PPM in global climate model dynamic cores as well as by Colella.  The author used this approach in his extrema-preserving methodology where the edge values are utilized to produce useful approximations in regions with smooth extrema.  The bottom line is that this approach has proven to be enormously flexible in practice due to the choice of the edge values coupled with a cell-average preserving reconstructing polynomial.  It is reasonable to have faith in the ability for other schemes presented next to be used in a similarly flexible manner.  Indeed the construction of approximations via polynomial reconstruction is highly amenable to such extensions as the geometric picture offered greatly aids the subsequent method development and testing.

One route to nonlinear stability are ENO methods that offer adaptive stencil selection.  These methods are based upon the choice of the “smoothest” stencil defined in some sense.  Another maxim from the ENO literature is the need to bias the stencils toward being upstream-centered.  Put differently, Van Leer’s paper presents a set of upstream-centered approximation that should be favored over other approximations.  Given this approach we might define an ENO scheme based on Scheme V as predominantly using that scheme, but choosing other more upwind or downwind stencils if the function being approximated is insufficiently smooth.

For a positive characteristic velocity an upwind stencil could define a parabola, by u^n_{j-1}, u^n_{j-1/2} and u^n_{j}.  This gives a parabola of

P(\xi) = u_j^n + (-\frac{5}{4}u_j^n - \frac{1}{4} u_{j-1}^n + \frac{3}{2} u^n_{j+1/2})\xi + (-\frac{9}{4} u_j^n + \frac{3}{4} u_{j-1}^n + \frac{3}{2} u_{j+1/2}^n)(\xi^2 -\frac{1}{12}).

This produces a semi-discrete scheme with third-order accuracy and a dispersion relation of \imath \theta + \frac{1}{18} \theta^4 + \frac{1}{270}\imath\theta^5+ O(\theta^6) .

A second stencil to choose from resulting in a third-order method is defined by u^n_{j-1}, u^n_{j+1/2} and u^n_{j}. This gives a parabola of

P(\xi) = u_j^n + (\frac{5}{2}u_j^n + \frac{1}{2} u_{j-1}^n - 3 u^n_{j-1/2})\xi + (\frac{3}{2} u_j^n + \frac{3}{2} u_{j-1}^n - 3 u_{j-1/2}^n)(\xi^2 -\frac{1}{12}).

This produces a semi-discrete scheme with third-order accuracy and a dispersion relation of \imath \theta - \frac{5}{72} \theta^4 + \frac{1}{270}\imath\theta^5+ O(\theta^6) .

The downwind stencil defined by  u^n_{j+1}, u^n_{j+1/2} and u^n_{j} ends up not being useful because it results in a scheme with centered support.  This gives a parabola of

P(\xi) = u_j^n + (3 u^n_{j-1/2}-\frac{5}{2}u_j^n + \frac{1}{2} u_{j+1}^n)\xi + (\frac{3}{2} u_j^n + \frac{3}{2} u_{j+1}^n - 3 u_{j-1/2}^n)(\xi^2 -\frac{1}{12}).

The updates become effectively decoupled because of cancelation of terms and the order of accuracy drops to second-order.  This produces a semi-discrete scheme with third-order accuracy and a dispersion relation of \imath \theta - \frac{1}{24} \imath\theta^3 + \frac{1}{1920}\imath\theta^5+ O(\theta^7) .

The same thing happens for a stencil of latex u^n_{j+1}$, u^n_{j-1/2} and u^n_{j}.

Now we can move to describing the performance of high-order methods based on Van Leer’s ideas, but extended to slightly larger support.  All these methods will, at most, reference their nearest neighbor cells.  The first is rather obvious and gives fifth-order accuracy.  The reconstruction is defined by cell-averages, u_j^n, u_{j-1}^n, u_{j+1}^n and cell-edge values, u_{j-1/2}^n, u_{j+1/2}^n.  This defines a quartic polynomial,

P(\xi) = u_j^n + \frac{1}{8}(10 u_{j+1/2}^n-10 u_{j-1/2}^n +u_{j-1}^n - u_{j+1}^n)\xi

+ \frac{1}{8} (30 u_{j-1/2}^{n} + 30 u_{j+1/2}^{n} -u_{j-1}^{n}-58 u_j^n -u_{j+1}^{n}) (\xi^2-\frac{1}{12})

+ \frac{1}{2}(u_{j+1}^n-u_{j-1}^n - 2u_{j+1/2}^n+2u_{j-1/2}^n)\xi^3

+\frac{1}{12} (-30 u_{j-1/2}^n - 30u_{j+1/2}^n +5u_{j-1}^n+50 u_{j}^n +5u_{j+1}^n)(\xi^4-\frac{1}{80}).

This produces a semi-discrete scheme with fifth-order accuracy and a dispersion relation of \imath \theta + \frac{1}{900} \theta^6 + \frac{31}{31500}\imath\theta^7+ O(\theta^8) .  This is exceedingly accurate with the dispersion relation peaking at \theta=\pi with an error of only 2.5%.

The next method was discussed and recommended by Phil Roe at the JRV symposium and a conversation at another meeting.  It uses the cell-average u_j^n, the edge values u_{j\pm1/2}^n, and the edge-first-derivatives, s_{j\pm1/2}^n.It is also similar to the PQM method proposed for climate studies.  The PQM method is implemented much like PPM in that the edge variables are approximated in terms of the cell-centered degrees of freedom.  Here we will describe these variables as independent degrees of freedom.  The polynomial reconstruction defined by these conditions is

P(\xi) = u_j^n + \frac{1}{4}(6 u_{j+1/2}^n-6 u_{j-1/2}^n -s_{j-1/2}^n - s_{j+1/2}^n)\xi

+ \frac{1}{4} (3 s_{j-1/2}^{n} - 3 s_{j+1/2}^{n} +30_{j-1/2}^{n}-60 u_j^n +30u_{j+1/2}^{n}) (\xi^2-\frac{1}{12})

+ (s_{j+1/2}^n+s_{j-1/2}^n - 2u_{j+1/2}^n+2u_{j-1/2}^n)\xi^3

+\frac{1}{2} (-30 u_{j-1/2}^n - 30u_{j+1/2}^n +5 s_{j-1/2}^n+60 u_{j}^n +5 s_{j+1/2}^n)(\xi^4-\frac{1}{80}).

This produces a semi-discrete scheme with fifth-order accuracy and a dispersion relation of \imath \theta + \frac{1}{7200} \theta^6 + \frac{1}{42000}\imath\theta^7+ O(\theta^8) .  This is exceedingly accurate with the dispersion relation peaking at $latex\theta=\pi$ with an error of only 1.2%.

We will close with a couple of schemes that are more speculative, but use the degrees of freedom differently than before.  Both will be based on a cubic polynomial reconstruction and involve the first moment of the solution along with either edge-values or edge-slopes.  Aside from the use of the first moment it is similar to the schemes above.  The reconstruction for the moment, cell-average and cell-edge values is

P(\xi) = u_j^n + \frac{3}{2}(20 m_{1,j}^n- u_{j+1/2}^n- u_{j-1/2}^n)\xi

\frac{1}{2}(u_{j-1}^n+ u_{j+1}^n - 2 u_j^n)(\xi^2 -\frac{1}{12}) .

+ (10 u_{j+1/2}^n-10 u_{j-1/2}^n - 120 m_{1,j}^n)\xi^3 .

As previous experience would indicate the use of the moment has remarkable properties, with this method producing the same error as the quartic reconstruction based on edge-values and slopes plus the cell-average, but one order lower reconstruction.

I will close by presenting three schemes based on cubic reconstructions, but utilizing the first moments.  Two will achieve fifth-order accuracy, and the last will suffer from cancellation effects that lower the accuracy to fourth-order.  These methods are unlike the discontinuous Galerkin methods in that their support will go beyond a single cell.  The first of the three will use the three cell-centered variables, and the moment of the central cell , giving a polynomial,

P(\xi) = u_j^n+\frac{3}{44}(200 m_{1,j}^n - u_{j+1}^n + u_{j-1}^n)

3(u_{j-1/2}^n+ u_{j+1/2}^n - 2 u_j^n)(\xi^2 -\frac{1}{12})

+ \frac{5}{11} (u_{j+1}^n - u_{j-1}^n - 24 m_{1,j}^n) \xi^3 .

This produces a semi-discrete scheme with fifth-order accuracy and a dispersion relation of \imath \theta + \frac{11}{7200} \theta^6 + \frac{1157}{3024000}\imath\theta^7+ O(\theta^8) .  This is exceedingly accurate with the dispersion relation peaking at $latex\theta=\pi$ with an error of only 3.1%.

The next method uses the cell-center values of the cell and it upwind neighbor and the moments in each of those cells.  The reconstructing polynomial is

P(\xi) = u_j^n+\frac{1}{8}(78 m_{1,j}^n -18 m_{1,j-1}^n -+3 u_{j}^n -3 u_{j-1}^n)

\frac{1}{4}(66 m_{1,j-1}^n+114 m_{1,j}^n -15 u_j^n+15 u_{j-1}^n)(\xi^2 -\frac{1}{12})

+ \frac{1}{2} (5 u_{j-1}^n - 5 u_{j}^n + 15 m_{1,j}^n +15 m_{1,j-1}^n) \xi^3 .

This produces a semi-discrete scheme with fifth-order accuracy and a dispersion relation of \imath \theta - \frac{1}{1350} \theta^6 + \frac{1}{23625}\imath\theta^7+ O(\theta^8) .  This is  accurate with the dispersion relation peaking at $latex\theta=\pi$ with an error about 12%.

Our final method uses the cell-centered value of the cell and its downwind neighbor along with the first moment in each of those cells.  The reconstruction takes the form,

P(\xi) = u_j^n+\frac{1}{8}(78 m_{1,j}^n -18 m_{1,j+1}^n -+3 u_{j+1}^n -3 u_{j}^n)

\frac{1}{4}(-66 m_{1,j+1}^n+114 m_{1,j}^n -15 u_j^n+15 u_{j+1}^n)(\xi^2 -\frac{1}{12})

+ \frac{1}{2} (5 u_{j}^n - 5 u_{j+1}^n + 15 m_{1,j}^n +15 m_{1,j+1}^n) \xi^3 .

We note that this stencil produces cancellation that lowers its accuracy by one order.  This produces a semi-discrete scheme with fifth-order accuracy and a dispersion relation of \imath \theta - \frac{1}{1080} \imath\theta^5 + \frac{1}{54432}\imath\theta^7+ O(\theta^8) .  This is  accurate with the dispersion relation peaking at $latex\theta=\pi$ with an error of 10%. The last two methods may not be particularly interesting in and of themselves, but may be useful as alternative stencils for ENO-type methods.

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