The document summarizes an edge-based finite-volume scheme for solving partial differential equations on triangular grids that achieves third-order accuracy without using curved elements. It presents a new boundary closure formula that allows the scheme to maintain third-order accuracy at boundary nodes. Numerical results are shown for the Burgers' equation and advection-diffusion equation on irregular triangular grids, demonstrating third-order convergence of errors at both interior and boundary nodes using the new boundary formula.

1. Accuracy-Preserving Boundary Quadrature
for Edge-Based Finite-Volume Scheme:
Third-order accuracy without curved elements
Hiroaki Nishikawa
National Institute of Aerospace
!
!
56th NIA CFD Seminar
December 16, 2014
JCP2015, v281, pp518-555

2. “You Already Have It.”
You already have what you want:
happiness, jobs, money, or anything.
The issue is always just how to
manifest what you already have.
The first step is to believe it.

3. Edge-Based Discretization

4. Edge-Based Finite-Volume Method
Edge-based finite-volume scheme:
with the upwind flux at edge midpoint:
NASA’s FUN3D; Software Cradle’s SC/Tetra; DLR Tau code, etc.
k
j
nr
jk
nℓ
jk
- 1st-order with nodal values
- 2nd-order with linear extrapolation, linear LSQ
- 3rd-order with linear extrapolation, quadratic LSQ Katz&Sankaran(JCP2011)
Accuracy with left/right states:
Efficient 3rd-order scheme: edge-loop with a flux per edge

5. Why Third-Order? Part I
Zero dissipation for quadratic solution
Linear extrapolation with quadratic LSQ gradients:
For a quadratic solution, it gives
and therefore,
The same left and right states, but not exact.
k
j
nr
jk
nℓ
jk
Averaged flux is the source of error.
JCP2015, v281, pp518-555

6. Exact for quadratic fluxes
Linear flux extrapolation with quadratic LSQ gradients:
For a quadratic flux, it gives
and the edge-based discretization becomes
The same left and right fluxes, but not exact.
k
j
nr
jk
nℓ
jk
Exact for quadratic fluxes, and thus third-order accurate
True for arbitrary triangles/tetrahedra
Why Third-Order? Part II
JCP2015, v281, pp518-555

7. Lost with Exact Flux
If the flux is exact for quadratic fluxes (e.g., quadratic
extrapolation or kappa=0.5 with UMUSCL), we have
and the edge-based discretization becomes
k
j
nr
jk
nℓ
jk
JCP2015, v281, pp518-555
TE=O(h), and so DE=O(h^2)
3rd-order is lost if the flux is exact for quadratic fluxes.
DO NOT use quadratic extrapolation nor kappa=0.5 for fluxes.

8. Edge-Based Discretization
1st-Order
2nd-Order
3rd-Order
(Linear LSQ gradients)
(Quadratic LSQ gradients)
k
j
nr
jk
nℓ
jk

9. “You Already Have It.”
EB Scheme for Diffusion, Source, Unsteady
3rd-order EB scheme for applies to various equations.
- Diffusion (Laplace) [ JCP2014 ]
Preprint accepted for publication in Journal of Computational Physics, 2014.
where
F =
⎡
⎢
⎢
⎣
au − νp
−u/Tr − (y − yj) q/Tr
(x − xj) q/Tr
⎤
⎥
⎥
⎦ , G =
⎡
⎢
⎢
⎣
bu − νq
(y − yj) p/Tr
−u/Tr − (x − xj) p/Tr
⎤
⎥
⎥
⎦ ,
where Tr = (2
√
νπ)−2
as described in Ref.[18], and (xj, yj) denotes the location of a node at wh
is discretized. The third-order edge-based discretization is constructed based on this system a
Section 2.1. For the second-order scheme, the terms involving (x − xj) and (y − yj), which re
terms, are not used, and a point-integration has been used for the source terms as describe
Ref.[18]. Note that u satisfies the advection-diffusion equation (43), and also the gradients a
(p, q) = (∂xu, ∂yu) to the same order of accuracy as demonstrated in Ref.[18]. The nodal gradien
in the edge-based discretization, are computed in the same way as in the previous case excep
∇uj = (pj, qj) and avoid the gradient computation for u [18]. The resulting second- and third-or
finite-volume schemes are referred to as SchemeII(2nd) and SchemeII(3rd), respectively, as in
domain is a square as in the previous case. For this problem, we set (a, b) = (1.23, 0.12), and ν
Re = 10. The following exact solution [23] is specified everywhere on the boundary:
u(x, y) = cos(2πη) exp
−2π2
ν
1 +
√
1 + 4π2ν2
ξ ,
where ξ = ax + by, η = bx − ay. For the variables, (p, q), we specify the exact gradients on t
except on the bottom (y = 0), where the variable q, corresponding to the normal derivative, is co
numerical scheme solving the third equation for q; the boundary flux quadrature is then require
errors are, therefore, computed for the third equation. In this problem, we focus on the ac
boundary nodes for isotropic and anisotropic grids.
Steady Conservation Law
- Source term [ JCP2012 ]
Steady Conservation Law
- Unsteady [ CF2014 ]
Steady Conservation Law (implicit time integration)

10. Two Approaches
Pincock and Katz, JSC, v61, Issue2, pp454-476
1. Modify the target equation - extra equations
(so that, the same 3rd-order EB scheme can be applied.)
!
!
2. Modify the scheme - extra computational work
diffusion, source terms, and unsteady equations
1st/2nd/3rd-order Hyperbolic NS Schemes [AIAA2014-2091]
(4th-order viscous with cubic LSQ gradients)
3rd-order EB scheme has already been demonstrated for NS.

11. EB Discretization at Boundary

12. Boundary Closure
If BCs are imposed through numerical fluxes, the residual needs to be
closed by boundary contributions at boundary nodes.
2
3
5
4
j
nL
B nR
B
Boundary contributions must be defined such that
the overall discretization is exact for linear/quadratic fluxes:
It depends on the element type.

13. Second-Order Formulas
Available for triangles, quadrilaterals,
tetrahedra, hexahedra, prisms, pyramids.
See Appendix B. in AIAA2010-5093
Note: Some regularity is required for quadrilaterals, hexahedra, prisms, and pyramids.
Exact for linear fluxes

14. Second-Order for Triangles
Appendix B. in AIAA2010-5093
2
3
5
4
j
nL
B nR
B
Boundary condition is set in the right (ghost) state,
and let the numerical flux determine the boundary flux.
This formula has been known for decades (see, e.g., AIAA Paper 95-0348, 1995).
3rd-order formula is not known at this point….
Let’s see what we get with the 2nd-order boundary formula.
See, e.g., NASA-TM-2011-217181, AIAA2014-2923

15. x
y
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Terrible Results
Advection-diffusion problem
We need a formula for third-order.
2nd-order boundary formula doesn’t seem compatible with 3rd-order EB scheme.
x 0.01
Solution

16. A Formula for Third-Order?
How can I derive such a formula? Very difficult…..
2
3
5
4
j
nL
B nR
B
The overall discretization must be exact quadratic fluxes.

17. “You Already Have It.”
Free yourself and reset your mindset:
Forget about deriving the formula.
Believe it.
Behold what you have.

18. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

19. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

20. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

21. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

22. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

23. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.

24. 2
3
5
4
1
j
EB scheme is 3rd-order accurate.
This is a boundary stencil, and we have 3rd-order!
So, we don’t really need a boundary formula!

25. 2
3
5
4
b
j
EB scheme is 3rd-order accurate.
Node b can be used as a ghost node.

26. Expanding the 3rd-order EB scheme, we find
2
3
5
4
j
b
nL
B nR
B
Oh, we have a formula!

27. A General Formula
3rd-order with straight boundary edges.
2
3
5
4
j
b
nL
B nR
B
Note: the second terms (j5 and j2 terms) require linear extrapolation.
It preserves 1st/2nd/3rd-order accuracy at boundary nodes.

28. Remaks
!
• A general formula derived for tetrahedra (3D).
!
• Curved elements should not be used.
The EB discretization is 3rd-order on linear triangular elements.
Immediately applicable to existing grids (high-order grids not needed).
Other 3rd-order schemes on linear elements:
See JCP2015, v281, pp518-555
[ AIAA2001-2595 ]
[ IJNMF2007 ]
- 3rd-order fluctuation-splitting scheme
- 3rd-order LSQ scheme
- 3rd-order residual-distribution scheme [Mazaheri and Nishikawa 2015]
Attractive alternatives to high-order methods in applications
for which 3rd-order accuracy is sufficient.

29. 2
3
5
4
j
b
nL
B nR
B
Special Case I: Flat Boundary
This gives third-order accuracy.
General formula becomes

30. 2
3
5
4
j
b
nL
B nR
B
This gives third-order accuracy.
Special Case II: Flat Uniform Boundary

31. Boundary Formulas
- One-point formula (1st-order, 3rd-order for flat uniform boundary)
2
3
5
4
j
b
nL
B nR
B
- Two-point formula (2nd-order)
- General formula (3rd-order)

32. Results

33. Numerical Results
!
• Third-order results (see for 2nd-order results)
• Burgers, advection-diffusion, Laplace
• All equations solved as a steady conservation law
• 10 Irregular triangular grids: 1089 to 409,600 nodes.
• Residuals reduced by 10 orders in all cases.
• Errors measured at interior and boundary nodes
JCP2015

34. Burgers’ Equation
where and
- Weak condition at top boundary (outflow); exact solution imposed elsewhere.
- Errors measured at interior nodes and boundary nodes separately.
Exact solution:
x
y
0 1
0
1
3rd-order EB scheme applied in the steady conservation form.
[ JCP2012 ]

35. Burgers’ Equation
Interior Nodes
Error doesn’t propagate back into the domain.
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log
10
(L
1
errornormofu)
DE−I, IRRG, 3rd, Burgers
One point
Two point
General
Slope 1
Slope 2
Slope 3
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log
10
(L
1
TEnorm)
TE−I, IRRG, 3rd, Burgers
One point
Two point
General
Slope 1
Slope 2
Slope 3
Boundary Nodes
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log
10
(L
1
errornormofu)
DE−B, IRRG, 3rd, Burgers
One point
Two point
General
Slope 1
Slope 2
Slope 3
3
10
9
8
7
6
5
4
3
2
1
0
1
Log10
(L1
TEnorm)
TE−B

36. Advection-Diffusion
- Weak condition at bottom boundary: specify (u,p) and compute q(=uy).
the exact solution imposed elsewhere.
- Errors measured in q at interior and boundary nodes separately.
x
y
0 1
0
1
3rd-order EB scheme applied in the steady conservation form.
[ JCP2014 ]

37. Advection-Diffusion
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log10
(L1
errornormofq)
DE−I, IRRG, 3rd, Adv−Diff
One point
Two point
General
Slope 1
Slope 2
Slope 3
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log10
(L1
TEnormforthethirdequation)
TE−I, IRRG, 3rd, Adv−Diff
One point
Two point
General
Slope 1
Slope 2
Slope 3
Interior Nodes
Boundary formula affects both boundary and interior:
1st/2nd/3rd-order with one/two- and general formula.
3 2.5 2 1.5
10
9
8
7
6
5
4
3
2
1
0
1
Log
10
(h)
Log
10
(L
1
errornormofq)
DE−B, IRRG, 3rd, Adv−Diff
One point
Two point
General
Slope 1
Slope 2
Slope 3
3 2.5
10
9
8
7
6
5
4
3
2
1
0
1
L
Log
10
(L
1
TEnormforthethirdequation)
TE−B, IRR
Boundary Nodes
Isotropic Grids

38. Advection-Diffusion
Anisotropic Grids
x
y
0 1
0
1
x
y
0 1
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Flat uniform boundary grid:
- One-point formula: 3rd-order for flat uniform boundary
- Two-point formula: 2nd-order

39. Advection-Diffusion
Anisotropic Grids
x
Y-derivativeofuaty=0
0 0.5 1
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Galerkin(3rd)
SchemeII(3rd) One-point
SchemeII(3rd) Two-point
Exact
Re= 10
x
Y-derivativeofuaty=0
0.5 1
0.32
0.33
0.34
0.35
Galerkin(3rd)
SchemeII(3rd) One-point
SchemeII(3rd) Two-point
Exact
Third-order boundary formula is essential.

40. Curved Boundary Problem
Potential flow over a cylinder
Exact solution Fully irregular grid

41. Governing Equation
Laplace equation for the stream function
Third-order EB discretization is applied in the steady
conservation form.
Extra variables, p and q, correspond to the velocity components.
Third-order EB scheme produces 3rd-order accurate (u,v).
[ JCP2014 ]

42. The last equation approximates
Boundary Condition
Strong and weak conditions
1. Outer boundary: Specify the exact solution.
2. Cylinder (slip wall)
This is where the boundary flux is required.

43. Normal and TangentVectors
1. Linear Approximation
2. Quadratic Approximation
Quadratic interpolation over 3 nodes in the parameter space of edge-length, s.
NOTE: - This is reconstruction.
No analytical geometry is required.
- Geometrical singularities (e.g., corner)
would require a special treatment.

44. Linear Approximation
1.8 1.6 1.4 1.2 1
5.5
5
4.5
4
3.5
3
2.5
2
Log10
(h)
Log
10
(L
1
errornormofq)
DE−I, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
1.8 1.6 1.4 1.2 1
3.5
3
2.5
2
1.5
1
0.5
0
Log10
(h)
Log10
(L1
TEnormforthethirdequation)
TE−I, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
Interior Nodes
No boundary formula gives 3rd-order accuracy
Boundary Nodes
1.8 1.6 1.4 1.2 1
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Log10
(h)
Log10
(L1
errornormofq)
DE−B, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
1.8 1.6
2
1.5
1
0.5
0
0.5
1
1.5
Log10
(L1
TEnormforthethirdequation)
TE−B
Discretization error in the x-velocity (u)

45. 1.8 1.6 1.4 1.2 1
5.5
5
4.5
4
3.5
3
2.5
2
Log
10
(h)
Log
10
(L
1
errornormofq)
DE−I, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
1.8 1.6 1.4 1.2 1
3.5
3
2.5
2
1.5
1
0.5
0
Log
10
(h)
Log
10
(L
1
TEnormforthethirdequation)
TE−I, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
Quadratic Approximation
Interior Nodes Boundary Nodes
1.8 1.6 1.4 1.2 1
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Log
10
(h)
Log
10
(L
1
errornormofq)
DE−B, IRRG, 3rd, cylinder
One point
Two point
General
Slope 1
Slope 2
Slope 3
1.8 1.6
2
1.5
1
0.5
0
0.5
1
1.5
Log
10
(L
1
TEnormforthethirdequation)
TE−B,
Only the general formula gives 3rd-order accuracy.
Discretization error in the x-velocity (u)

46. Quadratic Approximation
General FormulaTwo-point formula
The quadrature formula has a significant impact on the solution.
Contours of the x-velocity (u)

47. Conclusions
See JCP2015, v281, pp518-555 for details.
- Edge-based scheme is 1st/2nd/3rd-order through boundary nodes.
!
-We already had a general boundary quadrature formula.
!
- Numerical flux should NOT be exact for quadratic fluxes
!
- 3rd-order accurate without curved elements.
!
- Accurate normal vectors required for 3rd-order (quadratic interpolation)
See AIAA2014-2091 for NS results.

48. Don’t worry,
if Santa Claus doesn’t bring you a present.
You already have it!
Happy Holidays!