ExtremumPreserving Limiters for MUSCL and PPM
Limiters are nonlinear hybridization techniques that are used to preserve positivity and monotonicity when numerically solving hyperbolic conservation laws. Unfortunately, the original methods suffer from the truncationerror being order accurate at all extrema despite the accuracy of the higherorder method [1, 2, 3, 4]. To remedy this problem, higherorder extensions were proposed that relied on elaborate analytic and geometric constructions [5, 6, 7, 8]. Since extremumpreserving limiters are applied only at extrema, additional computational cost is negligible. Therefore, extremumpreserving limiters ensure higherorder spatial accuracy while maintaining simplicity. This report presents higherorder limiting for computing van Leer slopes and adjusting parabolic profiles. This limiting preserves monotonicity and accuracy at smooth extrema, maintains stability in the presence of discontinuities and underresolved gradients, and is based on constraining the interpolated values at extrema (and only at extrema) by using nonlinear combinations of derivatives. The van Leer limiting can be done separately and implemented in MUSCL (Monotone Upstreamcentered Schemes for Conservation Laws) [2] or done in concert with the parabolic profile limiting and implemented in PPM (Piecewise Parabolic Method) [9, 10]. The extremumpreserving limiters elegantly fit into any algorithm which uses conventional limiting techniques. Limiters are outlined for scalar advection and nonlinear systems of conservation laws. This report also discusses the order correction to the pointvalued, cellcentered initial conditions that is necessary for implementing higherorder limiting. The material herein complements Colella and Sekora [11]. Lastly, there is no guarantee that extremumpreserving limiters preserve positivity. To ensure this property, one should combine the limiting with FCT (FluxCorrected Transport) [3].
1 Algorithms for Scalar Advection
Consider the following scalar equation in one spatial dimension:
(1) 
At timestep , the averagevalued, cellcentered quantity over a finite volume of length is:
(2) 
MUSCL/PPM are conservative finite volume methods that are used to compute :
(3) 
where is the average of a linear/parabolic interpolant over the interval swept out by the characteristics crossing the cell face at and is given by:
(4) 
where is the CFL number and is the linear/parabolic interpolant, such that .
There are three variations of Godunovtype methods for which limiters can be implemented:

van Leer Limiter in MUSCL

van Leer Limiter + Parabolic Profile Limiter in PPM

Parabolic Profile Limiter in PPM
Each of these algorithm variations are discussed below.
1.1 Muscl

van Leer limit the differences , giving order results. Apply the corresponding boundary conditions.

Use the van Leer limited differences to compute order differences:
(5) 
Employ piecewise linear reconstruction by computing spatially extrapolated facecentered values at the low and high (left and right) edges of cells:
(6)
1.2 Ppm

van Leer limit the differences , giving order results. Apply the corresponding boundary conditions.

Employ either or order piecewise parabolic reconstruction:

Method 1: use the van Leer limited differences and compute spatially extrapolated facecentered values at the low and high (left and right) edges of cells:
(7) (8) (9) (10) (11) 
Method 2: employ either or order piecewise parabolic reconstruction without using the van Leer limited differences in Step 1:
(12) (13) (14) It is important to note that when (centered difference) the two formulations for the piecewise parabolic reconstruction are identical.


Limit the parabolic profile .

Use the PPM predictor values to reconstruct the parabolic profile:
(15)
1.3 Update Solution for MUSCL or PPM

Compute fluxes:
(16) 
Use the divergence of the fluxes to update the solution:
(17) (18)
1.4 Conventional Limiters
To complete the specification of the MUSCL and PPM schemes, one defines the conventional limiters used to constrain the interpolated profiles within each cell.
Conventional van Leer Limiter
Given a sequence of averagevalued, cellcentered quantities, the conventional van Leer limiter proceeds with the following steps [2, 9, 10]:

Compute onesided and centered differences:
(19) (20) (21) 
Apply the conventional van Leer limiter:
(22) (23) (24)
One significant defect of this method is the clipping of extremum when . This clipping sets as a precautionary measure for suppressing spurious oscillations.
How one arrives at the formula for the conventional van Leer limiter can be understood by considering the following example. Assume that one is not at an extremum, , and . The value of at facecenter is approximated by:
(25) 
Clearly, and . Therefore:
(26) 
Conventional Parabolic Profile Limiter
Given the higherorder reconstruction of such that , the conventional parabolic profile limiter proceeds with the following steps [9, 10]:

Adjust according to the following cases:
(27) 
Reconstruct given the adjusted values for .
One significant defect of this method is the adjustments to make the reconstruction a monotone profile as a precautionary measure for suppressing spurious oscillations. However, this constraint is more restrictive than is required to preserve monotonicity [13].
The adjustments to are derived from the interpolation polynomial described in the original Piecewise Parabolic Method:
(28)  
(29)  
(30) 
where is the CFL number, which also corresponds to a dimensionless length scale. This scale is associated with each grid cell such that the left side of a cell is designated and the right side of a cell is designated . Differentiating with respect to gives:
(31) 
By evaluating at the left and right sides of the cell:
(32)  
(33) 
Maximize with respect to by setting the derivatives equal to zero and solving the resulting equations. One arrives at the following result:
(34) 
2 ExtremumPreserving Limiters
For smooth solutions away from extrema, the MUSCL scheme is order accurate for linear advection whereas the PPM is order accurate for linear advection and order accurate in the limit of vanishing CFL number. However, the monotonicity constraints at extrema reduce the truncation error to even at smooth extrema. This reduction in the overall accuracy of the method also introduces a nonsmooth component to the error. To eliminate this problem, one constructs a new limiting scheme at extrema.
The defect of the standard approach to limiting is most easily seen in the MUSCL limiter. Away from extrema, the magnitude of the slope is computed as the minimum of three undivided differences: the centered difference and twice the onesided differences. In smooth regions away from extrema, the centered difference and the onesided differences all approximate such that the minimum is always defined by the centered difference. At discontinuities, one of the onesided differences is typically much smaller than the other two differences. Therefore, this onesided difference is chosen because it leads to a reduction in the slope and suppresses oscillations. However, the idea behind this method fails at extrema because the derivative vanishes. Furthermore, the onesided differences have opposite signs and this nonconstant multiple bounds the centered difference. In the original van Leer and PPM limiters, the solution is to simply drop the order of the method to order.
In the approach used in this report, one changes the limiters at extrema, and only at extrema, by using comparisons of different estimates of the derivatives as a basis for whether to limit the underlying linear scheme. If the solution is smooth at the extremum, then all of the estimates of the derivative are comparable and the limiter leaves the underlying linear scheme unchanged. Discontinuities, underresolved gradients, and highwavenumber oscillations are detected either by one of the estimates of the derivative being much smaller than the others or by the various estimates of the derivatives changing sign. Either effect triggers a nontrivial limiting of the interpolating function in the cell and a resulting suppression of oscillations.
2.1 ExtremumPreserving van Leer Limiter
The extremumpreserving van Leer limiter parallels the conventional van Leer limiter:

Compute onesided and centered differences as well as onesided differences that are an additional spatial stepsize away:
(35) (36) (37) (38) (39) 
Test for extrema. An extremum is defined if the following condition is satisfied:
(40) 
If the above extremum condition is not satisfied, then limiting follows the conventional van Leer method:
(41) (42) (43) 
If the above extremum condition is satisfied, then:

Compute onesided and centered derivatives:
(44) (45) (46) 
Find the minimum derivative over the fivecells in question:
(47) (48) 
Apply the modified van Leer limiter at the extremum:
(49) (50) (51)

designates a derivative while designates a difference. Derivatives and differences are similar operators that can be transformed backandforth when one considers the relevant Taylor expansion and multiplies/divides each operator by factors of . is a constant that is independent of the mesh spacing and as , the extremumpreserving van Leer limiter reduces to the conventional van Leer limiter. For most calculations, .
How one arrives at the tighter bound of for peak height in the extremumpreserving van Leer limiter can be understood by considering the following example. Assume that one is at a local maximum such that and . The van Leer limiting condition bounds the derivative on the high (right) edge of the cell:
(52) 
When looking for a bound on from the low (left) edge of the cell, an extremum near implies that:
(53) 
Therefore, one arrives at the following bound on :
(54) 
These bounds are summarized in the following inequality:
(55) 
where the comes from the condition that is the nearest facecentered point to the cellcentered point for which one can unequivocally assert that .
2.2 ExtremumPreserving Parabolic Profile Limiter
If the piecewise parabolic reconstruction is done without using van Leer limited differences, then one has to include an additional step that limits extrema at cell faces such that lies between adjacent cell averages. Van Leer limiting automatically enforces this constraint.

Test for extremum at cell faces:
(56) If no extremum is found, then proceed without any adjustment to .

If an extremum is found, compute onesided and centered derivatives:
(57) (58) (59) 
Find the minimum difference with respect to that is greater than zero:
(60) (61) 
Adjust according to:
(62)
The above formulas are arrived at by considering the centered derivative:
(63) 
The spatial resolution of enters because one is considering differences between . By substituting in the following expressions:
(64)  
(65) 
which convert pointvalued to averagevalued quantities, one arrives at:
(66) 
Rearranging the above equation gives the expression:
(67) 
After finding the minimum derivative , one can again use the above equation to assign a value to :
(68) 
Lastly, and one proceeds with limiting .
Now given the higherorder reconstruction of such that , adjust according to the following cases:


Compute the derivative with respect to from the the interpolation polynomial described in the original Piecewise Parabolic Method:
(69) It is important to note that this derivative with respect to is also a order difference for . Furthermore, this difference represents the maximum difference that can occur across a cell given a parabolic profile.

Compute onesided and centered order derivatives:
(70) (71) (72) 
Find the minimum difference that is greater than zero, given the derivatives over the fivecells in question as well as the above difference that was derived from the interpolation polynomial:
(73) (74) 
Adjust according to and :
(75)



Compute the maximum value of over a given cell:
(76) This formula is arrived at by averaging the interpolation polynomial from the original Piecewise Parabolic Method on the left side of cell :
(77) (78) (79) Maximize with respect to :
(80) Therefore:
(81) 
Preserve monotonicity by ensuring that the following condition is just satisfied:
(82) 
Solve the above equation for :
(83) (84)



Compute the maximum value of over a given cell:
(85) This formula is arrived at by averaging the interpolation polynomial from the original Piecewise Parabolic Method on the right side of cell :
(86) (87) (88) Maximize with respect to :
(89) Therefore:
(90) 
Preserve monotonicity by ensuring that the following condition is just satisfied:
(91) 
Solve the above equation for :
(92) (93)


Reconstruct given the adjusted values for .
3 Numerical Results
Results are presented for 1D scalar advection. The standard test problems were employed to demonstrate improvements in accuracy [14]. The following parameters were used in all test problems:
(94) 
Periodic boundary conditions were used with the following number of ghost cells:
Order Reconstruction  (95)  
Order Reconstruction  (96) 
The test problems were defined by the following initial conditions:
(97)  
(98)  
(99) 
The conventional initialization for order accurate methods is obtained by approximating the average over a cell by the value at the center of the cell, i.e., the midpoint rule for integrals. However, that initialization is only order accurate, and less accurate than what one expects for PPM when applied to the advection equation. For that reason, one uses a order accurate approximation to the cell average, following [12]:
(100) 
This order correction is only administered when defining the initial conditions. The corresponding boundary conditions are applied and one begins advancing the temporal loop. For smooth solutions away from extrema, PPM is order accurate for linear advection and order accurate in the limit of vanishing CFL number . The following definitions for the nnorms and convergence rates are used throughout this note. Given the numerical solution at resolution and the analytic solution , the error at a given point is:
(101) 
The 1norm and norm of the error are:
(102) 
The convergence rate is measured using Richardson extrapolation:
(103) 