International Journal of
Chemical Reactor Engineering
Volume 

Article A
Numerical Convection
Algorithms and Their Role in
Eulerian CFD Reactor
Simulations
Hugo A. Jakobsen∗
∗ Norwegian
University of Science and Technology, jakobsen@chembio.ntnu.no
c
Copyright 2003
by the authors.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the publisher, bepress.
Numerical Convection
Algorithms and Their Role in
Eulerian CFD Reactor
Simulations
Hugo A. Jakobsen
Abstract
In this paper a comparative convection algorithm study is presented. The
performance of a large number of schemes is compared evaluating the predicted
solutions for a standard benchmarking test problem. The nature of the errors
caused by the numerical approximations to the convection term is highlighted.
Although there is no algorithm that performs the best in general, several conclusions can be made. The tests performed show that the 1st order upwind scheme
and several variations of this scheme are very diffusive and should be avoided.
Most stable 2nd order schemes seem to be much more accurate, whereas the accuracy gained by higher order schemes (3rd order and 4th order) may be a little
more costly. Implicit time integration schemes are usually not as efficient as the
corresponding explicit schemes due to the computational time required on the iterative process. With larger time steps the accuracy of implicit schemes decrease
rapidly. The choice of proper higher order schemes (2nd order schemes) is then
seemingly determined by the trade−off between accuracy and computational
time. The conservative methods like the UTOPIA, the QUICK−1D combined
with a limiter, and a limited number of FCT and TVD formulations may be
sufficient solving the multi−fluid model equations. For advective terms (e.g.,
as occur in the temperature equation) the non−flux−based modified method
of characteristics is very fast, but also other higher order (2nd order) schemes
performed
KEYWORDS: multiphase reactors, Eulerian models, numerical diffusion, dynamic flow patterns, numerical methods, convection
Jakobsen: Numerical Convection Schemes
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1. INTRODUCTION
In the last two decades an increasing trend in applying computational fluid dynamics (CFD) to
elucidate details of the reactor performance has been seen in the literature and recognized as very useful
by the industry. Kuipers and van Swaaij (1997) provided a survey on the application of CFD to the field
of chemical reaction engineering.
Recent experimental studies on the flow structures of multiphase chemical reactors like bubble
column, fluidized bed- and stirred tank reactors have provided insight and evidence of the dynamic nature
of these systems. The instantaneous flow structures found in these reactors, are different from those
inferred by utilizing time average data. Steady-state model computations can thus not provide a rational
basis for the fundamental description of the interfacial mass, momentum, and energy transport processes.
The transient multiphase flow models may apparently more realistically describe the multiphase flow
structure. Furthermore, due to the relatively high holdup of the dispersed phases in operating reactors, the
Eulerian modeling framework has to be adopted.
In this paper we focus on an important aspect of dynamic Eulerian models, namely the errors
caused by the numerical approximations to the convection terms. Very different numerical properties are
built into the various numerical schemes proposed for solving the model equations of this type. Care has
to be taken to make sure that the numerical algorithm chosen is consistent with and reflects the actual
physics expressed by the theoretical model equations applied. The implementation of low accuracy
convection schemes may totally destroy the physics reflected by the sophisticated multiphase CFD model
formulations in use today (a typical two-fluid reactor model is given in appendix A).
The objective of investigating this problem is to gain insight into the expected errors and the
applicability of dynamic Eulerian methods to the CFD modeling of multiphase reactors. An ideal scheme
should satisfy several criteria: (1) positiveness, (2) conservativeness, (3) shape preservation, (4) small
numerical diffusion and dispersion, (5) accurate phase speed, (6) boundedness, (7) transportiveness, (8)
monotonicity, (9) entropy-satisfying, (10) accurate implementation of the boundary conditions, (11)
accurate resolution of discontinuities, (12) low computational costs, (13) low complexity (easy to
implement), (14) efficient parallelization, and (15) generality. These numerical properties are not entirely
independent, we merely want to highlight their importance. There is no single scheme that fulfil all these
criteria completely, however many methods may meet some of these requirements. In practice, it is still
an open question whether or not all of the listed properties could strictly be met by a single scheme.
It this paper we compare the performance of several schemes considered good candidates for use
in multiphase Eulerian reactor models. Most commercial multiphase CFD codes basically seeking steady
state solutions still resort to the classical upwind (or donor cell) method due to it's stability properties. For
many years the 3nd order QUICK scheme was considered favorable in single phase CFD because of the
improved accuracy obtained by this scheme. Lately, certain Flux-Corrected-Transport (FCT) schemes and
Total Variation Diminishing (TVD) schemes are claimed to be preferable in CFD. Other non-linear flux
limiters have also been developed intending to improve the performance of the basic schemes.
2. THEORETICAL ASPECTS
The equation describing the advection of a scalar variable, φ, yields
∂φ
(1)
+ v ⋅ ∇φ = 0
∂t
The conservative form of the above equation is derived by use of the continuity equation
∂ ( ρφ )
(2)
+ ∇ ⋅ ( ρ vφ ) = 0
∂t
where φ denotes the scalar variable transported, v denotes the fluid velocity vector and ρ denotes the fluid
density.
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The origin of the numerical errors involved solving the Eulerian model formulations is related to
the discretization problem and the choice of approximations to the differential equations. Numerical
methods constructed based on this advective form of the transport terms are shape preserving, but not
conservative. Schemes constructed based on the conservative form (or flux form) of the transport terms
are preferable when the model expresses a local conservation law for a conservative variable (i.e., in
contrast to the temperature equation that should be solved on the non-conservation form). These flux
based methods guarantee conservation of the transported variable φ (Roache, 1992), but are usually not
shape preserving. Thuburn (1995) and Leonard, Lock and MacVean (1996) discuss these numerical issues
in further detail.
In multiphase flow calculations implicit upstream differencing is still a commonly used method
for the convective terms in spite of the well-known and serious accuracy problems associated with the
implicit artificial viscosity of the method. According to Roache (1992), a simple Taylor series analysis on
the 1D transport equation shows that the transient artificial viscosity coefficients for explicit upwind
differencing is given by
w∆z
υ numerical =
(1 − CFL )
(3)
2
the corresponding implicit method gives
υ numerical =
w∆z
2
(1 + CFL )
(4)
where νnumerical denotes the numerical or artificial viscosity, w denotes the z-component of the velocity
vector, and CFL denotes the Courant number (based on the Courant-Friedrichs-Lewy condition).
It can be noted that at least the explicit upwind method for the constant velocity model gives the
exact answer for CFL=1, whereas the implicit upwind differencing method never does. The numerical
viscosity of the implicit method may increase a lot for CFL >> 1, which is the argument that does not
justify its use compared to explicit upwind differencing. This finding is the reason why we have primarily
included explicit methods in our test program.
The truncation error of advection and convetion schemes can be analyzed using the modified
equation method (Warming and Hyett, 1974). The presence of ∆z (i.e. the grid spacing) in the leading
error term indicates the order of accuracy of the scheme. The even-ordered derivatives in the error
represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed)
error. Artificial diffusion is thus built into all 1st order upstream schemes. The exception is the special
case when the Courant number is equal to 1, then the error term vanishes. Oscillations are produced if an
odd-order derivative gives a weighty contribution to the truncation error of the scheme. Even order
upwind methods tend to produce oscillations upwind of a change in gradient, while even order central
difference methods give oscillations downwind of a change in gradient. Another method for analyzing the
truncation error and the numerical stability properties of the schemes is the Fourier (or von Neumann)
method (e.g. O'Brien, Hyman and Kaplan, 1951; Odman, 1997).
Odman (1997) stated that all the 1st order upwind based schemes introduce some numerical
diffusion, so methods with comparatively low numerical diffusion should be preferred. Quantifying this
inherent numerical diffusion is desirable for at least two reasons. First, a quantitative comparison is very
helpful for the algorithm selection process. Knowing the amount of numerical diffusion facilitates the
choice from the pool of available algorithms. Second, the models explicitly account for the parameterized
diffusion. However, when predicting the physical diffusion, the contribution of inherent numerical
diffusion must be considered. An accurate estimate of inherent numerical diffusion is thus necessary to
control the total amount of diffusion in the model. If the inherent numerical diffusion completely
dominate the explicit parameterized diffusion, the numerical scheme applied is by no means
recommended.
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Jakobsen: Numerical Convection Schemes
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3. NUMERICAL SCHEMES
A large number of numerical discretization schemes approximating the convection terms for
scalar variables have been analyzed in this study. The performance of the various schemes is compared
evaluating the predicted numerical solution of a dynamic 2D-test problem having a known analytical
solution (see appendix B for further details). This numerical problem devised by Smolarkiewicz (1983),
determining the advection or convection of a scalar field in a fluid rotating at a constant angular velocity,
has been used as a numerical benchmarking test in many papers.
The design of numerical convection schemes is based on one of the three basic techniques
(Hirsch, 1988; Roache, 1998): Finite difference -, b) Finite volume -, and c) Weighted Residual Methods
(e.g., Finite Element-. Collocation-, Spectral- and Pseudospectral methods). The finite difference
methods are usually applied solving simple geometry problems formulated in Cartesian Coordinates. The
finite volume method is very often applied in fluid dynamics because of the inherent conservative
property. The weighted residual methods are usually not conservative and are not so often used in fluid
dynamics. These methods are better suited for structure problem analysis.
Furthermore, not all the schemes that are discussed here can be used solving both the momentum
vector equation (i.e., for negative velocities) as well as the transport equations for scalar variables, since
some of the methods intended for scalar variables include limiters that merely filter out negative values.
The performance of several finite differences and flux schemes considered good candidates for
use in multiphase Eulerian reactor models has been evaluated. The first explicit upwind scheme was
introduced by Courant, Isaacson and Reeves (1952), and later on several extensions to higher order
accuracy and implicit time integrations have been developed. Gudunov (1959) further developed the 1st
order upwind method introducing the concept of flux integral splitting methods. This concept determines
the basis for many modern higher order upwind methods. These methods involve the use of some sort of
exact or approximate Riemann solvers. Gudunov (1959) also showed that linear monotone schemes have
an order accuracy of at most 1. This order barrier does not apply to non-linear discretizations. Liou and
Edwards (1999) discussed several flux difference and flux vector splitting methods, emphasizing the
recent efforts to extend the Advection Upstream Splitting Method (AUSM) schemes to deal with low
Mach and multiphase flows. Three variations of the QUICK (Quadratic Upstream Interpolation for
Convective Kinetics) scheme of Leonard (1979) (i.e., QUICK 1D, QUICK-2D and QUICKEST (QUICK
with Estimated Streaming Terms) schemes) have been included in this test program. Note that the explicit
QUICK-1D and QUICK-2D schemes are unstable in the absence of diffusion. The UTOPIA scheme
(Uniformly Third-Order Polynomial Interpolation Algorithm) of Leonard, MacVean and Lock (1995) is
also considered. The UTOPIA and QUICK based algorithms were also combined with three different flux
limiters. These limiters stem from the Universal limiter (i.e., the associated monotonicity criteria)
presented by Leonard (1979). Leonard and co-workers introduced both the ULTIMATE (Universal
Limiter for Transient Interpolation Modeling of the Advective Transport Equations) strategy (Leonard,
1988) and later the ULTRA (Universal Limiter for Tight Resolution and Accuracy) approach (Leonard
and Mokhtari, 1990) to guarantee monotonicity. The performance of the ULTRA limiter has been
evaluated. Thuburn (1996) extended the 1D ULTIMATE limiter to unsteady advection, and a multidimensional strategy has later developed by Thuburn (1996) and Thuburn (1997). Both the 1D and the
multi-dimensional limiters of Thuburn are included in this work. Another approach aims at preventing the
generation of numerical oscillations, instead of damping them after they have been allowed. This
approach is based on the concepts of non-linear limiters introduced by van Leer (1974) and Boris and
Book (1973). The work of Boris and Book (1973) determines the basis for a group of methods called flux
correction transport (FCT) schemes. The work of van Leer (1974) and van Leer (1979), on the other hand,
represents an extension of the ideas of Gudunov to higher order accuracy. These approaches were later
generalized via the concept of total variation diminishing (TVD) schemes, introduced by Harten (1983),
whereby the variation of the numerical solution is controlled in a non-linear way, such as to forbid the
appearance of any new extremum. Such methods give higher order accuracy without dispersive ripples. A
common feature of most TVD schemes is that even the high order accurate schemes reduce to first order
at local extrema. It is noticed that the difference between the limiting processes of TVD and FCT schemes
lies in that the TVD schemes usually are of one step, while the FCT is of two steps. For further studies on
the FCT and TVD schemes the interested reader is referred to the textbook of Hirch (1990) and the
original papers.
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Four different TVD limiter formulations have been evaluated in this work. The limiters included
in this test are: a) the monotonic limiter of van Leer (1974): ψ (r) =(r+|r|)/(1+r), b) Minmod limiter
reported by Sweby (1984): ψ (r) =minmod(r,1), c) the SUPERBEE limiter of Roe (1985) and Roe (1986):
ψ (r) = max[0, min(2r,1), min(r,2)], d) the MUSCL (Monotonic Upwind Scheme for Convective Laws) of
van Leer (1979): ψ (r) = max[0,min(2,2r,(1+r)/2)]. These TVD limiters are functions of smoothness
monitors determining the local gradient of the variable field. Three monitors suggested by van Leer
(1974), Roe (1986) and Le Veque (1992) have been used in this study: a) ri+1/2 = (ϕi+1−σ−ϕi−σ)/(ϕi+1−ϕi), b)
ri+1/2 = (|ui+1/2−σ|-∆t/∆x u2i+1/2−σ ) (ϕi+1−σ−ϕi−σ)/[( |ui+1/2|-∆t/∆x u2i+1/2 ) (ϕi+1−ϕi)], c) ri+1/2 = ui+1/2−σ
(ϕi+1−σ−ϕi−σ)/[ ui+1/2 (ϕi+1−ϕi)]. σ = sign (ui+1/2). A class of these schemes are constructed in the hybrid
form of added low-order, FL, and high-order, FH, flux approximations (Yang and Przekwas, 1992):
TVD
H
L
L
H
L
Fi +1/ 2 = ψ i +1/ 2 Fi +1/ 2 + (1 − ψ i +1/ 2 ) Fi +1/ 2 = Fi +1/ 2 + ψ i +1/ 2 ( Fi +1/ 2 − Fi +1/ 2 )
(5)
where FTVD indicates that the convective flux (F) has been approximated by a TVD discretization scheme,
FH denotes a higher order approximation of the convective flux, and FL denotes a lower order
approximation of the convective flux. The Ψ−variable denotes one of the given TVD limiters.
The TVD schemes analyzed in this paper are constructed combining the 1st order (and diffusive)
upwind scheme and the 2nd order (and dispersive (oscillating)) central difference scheme. The TVD
schemes are globally 2nd order accurate, but reduce to 1st order accuracy at local extrema of the solution.
The above flux is 1st order accurate in time.
In parallel to the model development in the CFD community, several alternative methods have
been developed in the geophysical sciences seeking dynamic solutions to the various forecast models. The
dynamic terms in these models are often solved applying time splitting-, fractional step - and operator
splitting methods. Traditional operator splitting techniques based on sequential one-dimensional updates
may contain several splitting errors. Among the schemes tested are the fractional step and time splitting
schemes, Strang splitting (Strang, 1968), the MACHO (Multidimensional Advective Conservative Hybrid
Operator) and COSMIC (Conservative Operator Splitting for Multidimensions with Inherent Consistancy)
schemes by Leonard, et al. (1996). In addition, a 2nd order accurate explicit two step method derived by
MacCormack (1969), from a combined space and time splitting discretization has been included in the
test. A large number of numerical advection algorithms were described and evaluated for the use in
atmospheric transport and chemistry models by Rood (1987), and Dabdub and Seinfeld (1994). The
transport equations for the species mixing ratios occurring in air pollution models are often solved using
FCT schemes. Two such schemes are included in the present test, the FCT schemes of Smolarkiewicz
(1983) and Bott (1989 a and b). A popular alternative, the SOM (Second Order Moments) scheme
developed by Russel and Lerner (1981) and extended by Prather (1986), is also included. This method is
based on the flux integral concept. In European weather forecast models the non-flux-based modified
methods of characteristics have been very popular as they are very fast. In this work the semi-Lagrangian
advection schemes of Bates and McDonald (1982), McDonald (1984) and McDonald (1987) have been
evaluated.
Four implicit methods were also implemented to evaluate the accuracy and time consumption of
implicit vs. explicit schemes. The implicit schemes involved are: a) the upwind scheme, b) the SUDS
(Skew Upstream Differencing Scheme) scheme of Raithby (1976), c) the QUICK scheme, and d) the
TVD scheme with the SUPERBEE limiter.
4. RESULTS AND DISCUSSION
In the engineering CFD community the numerical convection schemes have been considered
generic tools that can be applied solving mathematically similar equations. Since the complex multi-fluid
CFD models contain convection terms that are very similar to the ones found in single-phase models (see
appendix A), we have chosen to test the numerical schemes on a single-phase benchmarking problem that
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Jakobsen: Numerical Convection Schemes
5
is much simpler than the multi-fluid reactor models. The simple test problem has the important advantage
that it has an analytical solution. This benchmarking problem has also been used analyzing the
performance of new schemes presented in the literature on numerical methods (see references given in
section 3). It should, however, be noted that the conclusions drawn from this benchmark test is not
generally valid, as the performance of the methods to some extent is problem dependent. On the other
hand, these results may provide important guidelines for the choice of solution method for a wide range of
applications. In several reports on multiphase simulations it is stated that if a numerical method doesn’t
perform sufficiently solving a simple problem, it is not very likely that it will do so solving much more
complex problems. The importance of analyzing the numerical properties built into the convection
schemes intended for CFD multiphase chemical reactor simulations has been discussed in a large numbers
of papers (e.g., Sokolichin and Eigenberger, 1994; Lafaurie et al., 1994; Delnoij et al. 1997; Jakobsen et
al. 1997; Sokolichin and Eigenberger, 1999; Lathouwers, 1999; Jakobsen, 2000; Joshi, 2001).
With the above discussion in mind, we perform our numerical analyses on the benchmarking
problem intending to provide guidelines for the optimal choice of numerical convection schemes required
solving Eulerian reactor models.
The tests of the explicit methods have been performed with a time step of 1.0s, while the implicit
methods have been tested both with time steps of 1.0s and 4.0s. All the explicit methods were run both
with “single step” and “fractional step” node update.
Using the default upwind scheme a typical result is obtained as shown in figure 1 (left). The 1.
order upwind scheme is very diffusive and the predicted profile is not very accurate as can be seen
comparing the result with the analytical solution (figure 6, left). However, the profile predicted by the
QUICKEST scheme, figure 1 (right), indicates that there exist alternative algorithms that perform better
than the upwind scheme. Unfortunately, as mentioned earlier, there exist no general scheme that will
perform best for all applications. In this section we will thus focus on a few numerical properties that are
important solving Eulerian reactor models (e.g., time consumption, peak preservation, and absolute error).
To illustrate the differences in time consumption between the different methods, the CPU time
needed for advecting 1 revolution is shown in figure 2. The upstream scheme is the simplest to implement
and the fastest of the tested schemes, but this does not justify the prevailing diffusion caused by the
truncation error. The results of the Gudunov method are very close to those obtained by the explicit
upstream method. The MacCormack’s method is the second fastest after the first order upwind method.
Even though the MacCormack method has an attractive calculation speed, the magnitude of undershoots
and oscillations render the method not generally recommendable. Among the explicit schemes on their
“basic” form, the Bott-, SOM-, and Smolarkiewicz (with 10 corrections) schemes are the most CPUdemanding ones. The MACHO and COSMIC schemes are a factor of 10-20 more time consuming than
the simpler operator splitting schemes included in the test. For the low order schemes only negligible
improvements in accuracy are obtained using these techniques. Furthermore, a general trend is that the
implicit methods are slower than the explicit methods.
QUICKEST
4
4
3
3
2
2
1
1
φ
φ
Upstream
0
0
−1
−1
−2
−2
30
30
25
20
400*1.00s
15
10
400*1.00s
10
5
30
20
25
20
15
5
single step
25
30
25
20
15
15
10
5
no limiter
10
5
single step
Figure 1: Typical results. Predicted cone profiles using the upwind – (left) and the QUICKEST (right)
schemes.
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A graphical representation is made for the peak value preservation. Figure 3 shows the
percentage of the initial peak value after advecting 1 revolution. Figure 4 compares the average absolute
error of the different scheme combinations. A general trend is that the unstable explicit schemes, the
implicit methods, together with MacCormack and the first order methods have the highest average
absolute errors, and the flux-limited schemes the lowest. The implicit version of QUICK 1D is
unconditionally stable, but it is computationally inefficient. Increasing the size of the time step caused an
additional smearing of the cone. When using the explicit schemes with flux-limiters, the instability is
circumvented as undershoots are heavily reduced or eliminated, depending on the flux-limiter used.
When “single step update” is used, the results are best with the multidimensional limiter of Thuburn.
When combining the fractional step technique with any of the two one-dimensional limiters, impressive
results are obtained. However, the ULTRA limiter fails to completely eliminate undershoots. With
Thuburns one-dimensional limiter, no negative values are produced, and the average absolute errors are 5
times lower than for the upstream scheme. Considering the good peak preservation, low diffusion and the
absence of negative values when running the QUICK 1D scheme with fractional step update and
Thuburn’s one-dimensional limiter, this is found to be the most viable alternative. However, when it
comes to accuracy and peak preservation, the Bott and SOM schemes show impressive performance in
competition with the algorithms discussed above. On the other hand, the Bott- and SOM schemes are
respectively 4-5 and 9-10 times slower than the fractional step QUICK 1D with ULTRA or Thuburn 1D
limiter. The QUICKEST scheme has a convected profile visually comparable to that of UTOPIA, but it is
faster at the cost of accuracy and undershoots. Combining the method with flux-limiters
reduces/eliminates undershoots and reduces the average absolute error, but, in addition to the increase in
calculation time, it has a flattening effect on the peak.
The choice of smoothness monitors seems not to be very important for the performance of the
TVD schemes, whereas the choice of limiter is crucial. Among the 12 TVD schemes included in this test,
the SUPERBEE based scheme is recommended considering both accuracy and cpu time consumption.
The Skew Upstream Differencing Scheme (SUDS) suffers from an overwhelming CPU-time
usage and low accuracy. The diffusive effects in the flow direction are visually similar to the behavior
shown by the standard upstream scheme, but the transverse smearing is reduced. The implicit TVDSUPERBEE scheme gives the best results of the tested implicit schemes. However, compared to the
explicit schemes, the implicit TVD-SUPERBEE scheme shows a low peak preservation, high calculation
time requirement and relatively low accuracy. The scheme suffers badly from using the tested 4.0s
timestep size, reducing the peak to 14.6% and giving one of the highest values for the average absolute
error of all scheme configurations. It was shown that the implicit algorithms retained stability when the
time step (and thus the Courant number) was increased by four times in our test-environment. The
widespread use of implicit upstream (or QUICK) with Courant numbers ten- or even hundredfold the
magnitude of what is used in an explicit method, is not found justifiable in the presence of gradients or
steps in the convected variable. However, for transport of a nearly homogenous fluid, it offers a fast
solution.
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Figure 2. CPU time consumption transporting the cone one revolution is given for the various methods
included in the test program. Abbreviations used in the figure; fs: (Alternating) fractional step update, ss:
Single step update, imp: implicit update, long ts: Timestep size set to 4.0 s, Th1Dlim: Thuburn 1D flux
limiter, Th2Dlim: Thuburn multi-dimensional limiter, ULTRA: Leonard’s ULTRA approach,
UTOPIA+4th: 4th order contributions have been included in UTOPIA formulation, SUDS: Skew
Upstream Differencing Scheme.
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Figure 3. Peak preservation in %, obtained transporting the cone one revolution, as given for the various
methods included in the test program.
Abbreviations: fs: (Alternating) fractional step update, ss: Single step update, imp: implicit update, long
ts: Timestep size set to 4.0 s, Th1Dlim: Thuburn 1D flux limiter, Th2Dlim: Thuburn multi-dimensional
limiter, ULTRA: Leonard’s ULTRA approach, UTOPIA+4th: 4th order contributions have been included
in UTOPIA formulation, SUDS: Skew Upstream Differencing Scheme.
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Figure 4. Average absolute error occurring after transporting the cone one revolution as given for the
various methods included in the test program.
Abbreviations: fs: (Alternating) fractional step update, ss: Single step update, imp: implicit update, long
ts: Timestep size set to 4.0 s, Th1Dlim: Thuburn 1D flux limiter, Th2Dlim: Thuburn multi-dimensional
limiter, ULTRA: Leonard’s ULTRA approach, UTOPIA+4th: 4th order contributions have been included
in UTOPIA formulation, SUDS: Skew Upstream Differencing Scheme.
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When comparing the three evaluated flux-limiters, ULTRA has its main strength in speed. It has
the disadvantage that it cannot guarantee positiveness. The 1D limiter of Thuburn (1996) has the
advantage of guaranteeing local monotonic behavior and hence positiveness when used in a fractional
step update. Contrary to the ULTRA approach, it is designed to handle a spatial variation in the velocity
field, making it better suited for our test system. Its only disadvantage when being compared to the
ULTRA limiter is the higher time consumption. Of the tested flux-limiters, the Thuburn multidimensional limiter is the only one that can completely guarantee local monotonic behavior and
positiveness in an initially monotonic and positive region when the two-dimensional system is updated
with “single step”. While fast methods (like the three explicit QUICK variants) usually produce
oscillations and need flux limiters, the tripled overall calculation time might be too repulsive. On the
other hand, more sophisticated, higher order methods often do not show the same need for flux limiters,
even though the added calculation requirement is less significant. However, when small oscillations are
fatal and accuracy is of higher priority than calculation speed, the combination of an advanced flux
calculation scheme with this limiter would be a natural choice.
The box initial condition has been used as a verification and attestation of the conclusions drawn
from the cone test problem. The methods evaluated are: a) QUICK 1D alternating fractional step with
Thuburn 1D flux-limiter, b) UTOPIA single step, c) Bott, d) Explicit TVD-SUPERBEE. The UTOPIA
scheme suffers from a smearing of the profile, combined with over- and undershoot. Similarly, the Bott
scheme demonstrates overshoot and shape degeneration, but without negative values. The explicit TVDSuperbee scheme (figure 5, left) and QUICK 1D fractional step with Thuburn 1D flux-limiter (figure 5,
right) predicted the best results in this test (comparing the results with the analytical solution given in
figure 6, right).
QUICK 1D
4
4
3
3
2
2
1
1
φ
φ
TVD
0
0
−1
−1
−2
−2
30
30
25
20
400*1.00s
25
30
20
15
15
10
5
1 − Superbee limiter
400*1.00s
10
5
30
20
25
25
20
15
15
10
5
Thuburn 1D Universal limiter
10
5
alternating fractional step
Figure 5: Predicted box profiles . Left hand side figure: using the TVD (SUPERBEE) , Right hand side
figure: QUICK-1D combined with the Thuburn 1D flux limiter.
5. CONCLUSIONS
The tests performed show that the 1st order upwind scheme and several variations of this scheme
are very diffusive and should be avoided. Most stable 2nd order schemes seem to be much more accurate,
whereas the accuracy gained by higher order schemes (3rd order and 4th order) may be a little more costly.
Implicit time integration schemes are usually not as efficient as the corresponding explicit schemes due to
the computational time required on the iterative process. With larger time steps the accuracy of implicit
schemes decrease rapidly. The choice of proper higher order schemes (2nd order schemes) is then
seemingly determined by the trade-off between accuracy and computational time. The conservative
methods like the UTOPIA, the QUICK-1D combined with a limiter, and a limited number of FCT and
TVD formulations may be sufficient solving the multi-fluid model equations. For advective terms (e.g.,
as occur in the temperature equation) the non-flux-based modified method of characteristics is very fast,
but also other higher order (2nd order) schemes performed well.
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Jakobsen: Numerical Convection Schemes
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6. ACKNOWLEDGEMENTS
The author is grateful for receiving the original source codes for the MACHO, COSMIC,
QUICK and UTOPIA schemes from Leonard, B. P. and co-workers at the University of Akron, USA. The
source codes of the semi-Lagrangian schemes have been received from McDonald, A. at the Irish
Meteorological Service, Ireland. The source code of the SOM scheme has been provided by Sundet, J. at
UiO, and the source code of the Bott scheme has been obtained from MSC-W at DNMI in Oslo, Norway.
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Jakobsen: Numerical Convection Schemes
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8. APPENDIX A
The typical two-fluid reactor model equations are listed in this appendix (e.g., Jakobsen et al. 1997).
The generic local instantaneous equation for the k-th phase:
∂
( ρ kψ k ) + ∇ ⋅ ( ρ k v kψ k ) + ∇ ⋅ J k −
∂t
N
∑ρ
k ,c
φ k ,c = 0
(A1)
c =1
The first term denotes the time dependency of the generalized variable ψ, the second term denotes the
convection, the third term denotes the diffusion and the last term denotes bulk or volume sources.
The generic local jump condition:
2
∑ (( ρ ( v
k
k
− v I )ψ k + J k ) ⋅ n k ) = M I
(A2)
k =1
C o n s e rv e d
q u a n tity
ψ
D iffu s io n
flu x
J
V o lu m e tric
s o u rc e
φ
In te r fa c ia l
s o u rc e
MI
T o ta l m a s s
1
0
0
0
C om ponent
m ass
ωc
Jc
Rc
0
M o m e n tu m
v
T
gc
M Iσ
T⋅v +q
vc ⋅gc
ε Iσ
E n e rg y
e +
1 2
v
2
Table 1: Variables in the generic transport and jump equations.
Physical meaning of the variables:
ψ denotes the transported variable, J the diffusive fluxes, φ the volumetric sources,
MI interfacial sources, ωc the chemical component mass fraction, Rc the reaction rate,
v the velocity vector, T the total stress tensor, e the internal energy, q the heat flux, gc external
force fields like gravitation, MIσ and εIσ the surface tension forces and energy.
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9. APPENDIX B
The benchmarking problem is defined in this appendix (e.g., Smolarkiewicz, 1983).
We seek the solution of the two-dimensional advection equation for a transported scalar variable φ:
∂φ
∂φ
∂φ
− ωy
+ ωx
=0
∂t
∂x
∂y
(B1)
where the velocity profile (i.e., u=ωy, and v=ωx) is predefined as the angular velocity is given by:
2π
T
ω=
(B2)
The period of rotation, T, denotes the exact time in seconds required for a scalar (injected at a given point
in the velocity field) to be transported a full revolution around the center. This time span is set to 400s
throughout this work. Using a time step of ∆t = 1.0s, the exact convection would require 400 time steps
for one revolution. The spatial resolution is 32 x 32 grid cells in the xy-plane.
The initial conditions for the cone and box profiles are defined in equation (B3) and (B4) and shown in
figure (6).

(i − 8) 2 + ( j − 16) 2

φ (i, j ) = max( 0.0, 4 * 1 −

4


 )


0
(B3)
1.5 if (5 < i < 14) ∧ (12 < j < 20)
0.0 else
0
φ (i, j ) = 
(B4)
The exact solution to this two-dimensional advection problem can be found analytically. After one
complete rotation, the exact solution equals the initial profile.
Initial box
4
4
3
3
2
2
1
1
φ
φ
Initial cone
0
0
−1
−1
−2
−2
30
30
25
0*1.00s
25
30
20
20
15
15
10
5
10
30
20
25
0*1.00s
25
20
15
15
10
5
5
10
5
Figure 6: Initial conditions in the xy-plane for the cone (left) and the box (right) profiles. Profiles are also
equivalent to the exact solution after an integer number of rotations.
Finally, the predefined velocity distribution and the boundary conditions in the xy-plane are sketched in
figure 7.
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Jakobsen: Numerical Convection Schemes
15
35
30
25
20
15
10
5
0
-4
-4
0
5
10
15
20
25
30
35
External nodes set to zero
External nodes given the value of the closest outermost internal node
System boundary, outer side of the outermost internal nodes
Velocity vector of the center of a node
Figure 7: Predefined velocity distribution and the boundary conditions in the xy-plane.
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