High order direct discontinuous Galerkin method
Jue YanIowa State University
Lecture Series of High-Order Numerical MethodsJuly 27 – August 14, 2020
Outline
Introduction of direct DG method as a diffusion solver
Derivation of DDG method Numerical flux coefficients Relation to interior penalty DG (IPDG) method DDG scheme for nonlinear diffusion DDG method on triangular meshes
Advantages of DDG method
Maximum-principle-satisfying or positivity-preserving Super convergence Elliptic interface problem
Current and future research work Deep Neural Network as an ODE solver
DG method for hyperbolic type PDEs
0=+ xt uu
0ˆˆ 2/12/12/12/1 =−+− +−−
−++∫∫ jjjj
Ix
It vuvudxuvdxvu
jj
−++ = 2/12/1ˆ jj uu
−+ 2/1ju
++ 2/1ju
2/1+jx
DG formulation:
Numerical flux:
jI
discontinuous
2/1+jx2/1−jx
Direct DG (DDG) method for parabolic diffusion
+∆+∆++∆
=±
][][][ˆ 32102/1 xxxxxxxxx uxuxu
xuu
jβββ
2 ,][ :Note
−+−+ +
=−=uuuuuu
0=− xxt uuheat equation:
2/1at ? ±= jx xu−u +u
xj+3/2xj-1/2 xj+1/2
jI 1+jI
discontinuous
H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM Journal on Numerical Analysis, 47, No 1(2009), 675--698.
0)ˆ()ˆ( 2/12/12/12/1 =+−+ +−−
−++∫∫ jjxjjx
Ixx
It vuvudxvudxvu
jj
Identify numerical flux coefficients ][][ˆ 10 xxxx uxux
uu ∆++∆
= ββ
𝑢𝑢𝑥𝑥(𝑥𝑥𝑗𝑗−12
) ≈ 𝑑𝑑𝑑𝑑𝑥𝑥
𝑝𝑝(𝑥𝑥)𝑥𝑥𝑗𝑗−1/2
2/3−jx 2/1−jx**
2/1+jx
* *jI
1−jI
𝑢𝑢𝑗𝑗−1(𝑥𝑥) 𝑢𝑢𝑗𝑗(𝑥𝑥)polynomial 𝑝𝑝(𝑥𝑥)
Construct a cubic degree Lagrange interpolation polynomial
𝑝𝑝 𝑥𝑥 = 𝑢𝑢𝑗𝑗−3/2(𝑥𝑥−𝑥𝑥𝑗𝑗−1)(𝑥𝑥−𝑥𝑥𝑗𝑗)(𝑥𝑥−𝑥𝑥𝑗𝑗+1/2)
−3∆𝑥𝑥3/2+ 𝑢𝑢𝑗𝑗−1
(𝑥𝑥−𝑥𝑥𝑗𝑗−3/2)(𝑥𝑥−𝑥𝑥𝑗𝑗)(𝑥𝑥−𝑥𝑥𝑗𝑗+1/2)3∆𝑥𝑥3/4
+ 𝑢𝑢𝑗𝑗(𝑥𝑥−𝑥𝑥𝑗𝑗−3/2)(𝑥𝑥−𝑥𝑥𝑗𝑗−1)(𝑥𝑥−𝑥𝑥𝑗𝑗+1/2)
−3∆𝑥𝑥3/4+ 𝑢𝑢𝑗𝑗+1/2
(𝑥𝑥−𝑥𝑥𝑗𝑗−3/2)(𝑥𝑥−𝑥𝑥𝑗𝑗−1)(𝑥𝑥−𝑥𝑥𝑗𝑗)3∆𝑥𝑥3/2
𝑝𝑝′ 𝑥𝑥𝑥𝑥𝑗𝑗−1/2
=𝑢𝑢𝑗𝑗−3/2 − 8𝑢𝑢𝑗𝑗−1 + 8𝑢𝑢𝑗𝑗 − 𝑢𝑢𝑗𝑗+1/2
6∆𝑥𝑥
Carry out Taylor expansion for 𝑢𝑢𝑗𝑗−2/3, 𝑢𝑢𝑗𝑗−1 around 𝑢𝑢𝑗𝑗−1/2− and 𝑢𝑢𝑗𝑗 ,𝑢𝑢𝑗𝑗+1/2 around 𝑢𝑢𝑗𝑗−1/2
+
𝑝𝑝′ 𝑥𝑥𝑥𝑥𝑗𝑗−1/2
=1
6∆𝑥𝑥7 𝑢𝑢+ − 𝑢𝑢− + 3∆𝑥𝑥 𝑢𝑢𝑥𝑥+ + 𝑢𝑢𝑥𝑥− +
∆𝑥𝑥2
2𝑢𝑢𝑥𝑥𝑥𝑥+ − 𝑢𝑢𝑥𝑥𝑥𝑥− −
∆𝑥𝑥4
288𝑢𝑢4𝑥𝑥+ − 𝑢𝑢4𝑥𝑥− + ⋯
= 76
[𝑢𝑢]∆𝑥𝑥
+ 𝑢𝑢𝑥𝑥 + ∆𝑥𝑥12
𝑢𝑢𝑥𝑥𝑥𝑥 − ∆𝑥𝑥3
288𝑢𝑢4𝑥𝑥 + ⋯
Derivative at discontinuity (PDE point of view)
Riemann problem for heat equation:
∫∞
∞−
−−= dyyget
txu tyx )(41),( 4/)( 2
π
==−
)()0,(0
xgxuuu xxt
0=x
)(xg
++++=
=
′+−=
∂∂
∫∞
∞−
−−+
=
xxxxxx
ty
x
gtgtggt
dyygeggtx
u
][][41
)()0()0(41 4/
0
2
ππ
π
By the solution formula:
What is the derivative at the discontinuity: ∫∞
∞−
−==∂∂ dyyg
tye
ttx
xu ty )(
241),0( 4/2
π
2 ,][
−+−+ +
=−=ggggggNotation:
Comparison to the IPDG method
0 )]([21)]([
21 |ˆ 2/12/1
2/12/1 =++−+ −
++
−+−∫∫ jxjx
jjx
Ixx
It vuvuvudxvudxvu
jj
1. IPDG method (Arnold 1982):
2. Direct DG with Interface Correction (Yan and Liu, 2010):
0)]([21)]([
21][ 2/12/1
2/1
2/10
2/1
2/1=++
∆−−+ −
++
−+
−
+
−∫∫ jxjx
j
j
j
jxI
xxI
t vuvuvx
uvudxvudxvujj
β
][][ˆ 10 xxxx uxux
uu ∆++∆
= ββ
0=− xxt uu
DDG method degenerates to IPDG with constant and linear approximations
][][][ ][),(j
vx
uuvdxvuvuvuj
xI
xxj
x
j∆
+++= ∑∑ ∫∑ βBBilinear form:
),( vuB
Variations of DDG methods 0=− xxt uu
0)]([21)]([
21 |ˆ 2/12/1
2/12/1 =+++− +
−−+
+− ∫∫ jxjx
Ixx
jjx
It vuvudxvuvudxvu
jj
][][ˆ 10 xxxx uxux
uu ∆++∆
= ββ
DDG with interface correction (DDGIC) – suitable for parabolic PDE
Symmetric DDG method - more suitable for elliptic type PDE
Non-symmetric DDG method
0][~][~ |ˆ 2/12/12/12/1 =+++− −+
+− ∫∫ jxjx
Ixx
jjx
It uvuvdxvuvudxvu
jj
∆++∆
=
∆++∆
=
][][~
][][ˆ
10
10
xxxvx
xxxux
vxvx
vv
uxux
uu
ββ
ββ
0][~][~ |ˆ 2/12/12/12/1 =−−+− −+
+− ∫∫ jxjx
Ixx
jjx
It uvuvdxvuvudxvu
jj
DDG for nonlinear diffusion equations
0)( with 0))(( ≥=− uauuau xxt
∫=u
dssaub )()(
∆++∆
=
∆++∆
==
][][~
])([)()]([)()(
10
10
xxxvx
xxxuxx
vxvx
vv
ubxubxububuua
ββ
ββ
( ) 0)]([~)]([~ )( |)( 2/12/1
2/12/1 =+++−
−+∫∫ +− jj
jj
xxxxI
xxjjx
It ubvubvdxvuuavubdxvu σ
𝐼𝐼𝑗𝑗
(𝑎𝑎(𝑢𝑢)𝑢𝑢𝑥𝑥)𝑥𝑥 𝑣𝑣 𝑑𝑑𝑥𝑥 = 𝐼𝐼𝑗𝑗𝑏𝑏(𝑢𝑢)𝑥𝑥𝑥𝑥 𝑣𝑣 𝑑𝑑𝑥𝑥
= 𝑏𝑏(𝑢𝑢)𝑥𝑥𝑣𝑣𝑗𝑗+1/2− − 𝑏𝑏(𝑢𝑢)𝑥𝑥𝑣𝑣𝑗𝑗−1/2
+ − 𝐼𝐼𝑗𝑗𝑎𝑎(𝑢𝑢)𝑢𝑢𝑥𝑥𝑣𝑣𝑥𝑥 𝑑𝑑𝑥𝑥
Numerical flux:
0=∆− uut
DDGIC scheme:
Numerical flux:
DDG method on triangular meshes
𝐾𝐾𝑢𝑢𝑡𝑡𝑣𝑣 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑 −
𝜕𝜕𝐾𝐾𝑢𝑢𝒏𝒏𝑣𝑣 𝑑𝑑𝑠𝑠 +
𝐾𝐾∇𝑢𝑢 𝛻𝛻𝑣𝑣 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑 +
𝜕𝜕𝐾𝐾𝑣𝑣𝐧𝐧 [𝑢𝑢]𝑑𝑑𝑠𝑠 = 0
𝑢𝑢𝒏𝒏 = ∇𝑢𝑢 𝒏𝒏 = 𝛽𝛽0[𝑢𝑢]ℎ𝐾𝐾
+ 𝑢𝑢𝒏𝒏 +𝛽𝛽1 ℎ𝐾𝐾[𝑢𝑢𝒏𝒏𝒏𝒏]
𝑢𝑢𝒏𝒏𝒏𝒏= 𝛻𝛻𝑢𝑢𝒏𝒏 𝒏𝒏 = 𝛽𝛽0𝑢𝑢ℎ𝐾𝐾
𝑛𝑛12 + 𝑛𝑛22 + 𝑢𝑢𝑥𝑥𝑛𝑛1 + 𝑢𝑢𝑦𝑦𝑛𝑛2+𝛽𝛽1ℎ𝐾𝐾 𝑢𝑢𝑥𝑥𝑥𝑥 𝑛𝑛12 + 𝑢𝑢𝑦𝑦𝑥𝑥 𝑛𝑛1𝑛𝑛2 + 𝑢𝑢𝑥𝑥𝑦𝑦 𝑛𝑛1𝑛𝑛2 + 𝑢𝑢𝑦𝑦𝑦𝑦 𝑛𝑛22
𝒏𝒏=(𝑛𝑛1, 𝑛𝑛2)Outward normal vector:
0))(( =∇⋅∇− uuAut
Analysis on the numerical flux coefficients
+
−−
−+≥
4)
4)1(()
12)1((1 222
1
2222
10kkkkk ββ
γαβ
Theorem: the numerical flux is admissible provide that
To minimize the penalty parameter, we take and we have)1(4
32
*11 −==
kββ
41 2
0k
γαβ +≥
Proof: we need some combinatorial properties of the Hilbert matrix
1D heat equation with DDGIC scheme
Comparison to IPDG method: the penalty coefficient decrease from to a constant 2k 2
1D fully nonlinear equation
=
=−+−tx
xxxt
etxutxfuuu
),(),()()( 22
])[()(][)( 2
12
2
02
xxxx uxux
uu ∆++∆
= ββ
Remark: super convergence (k+2)th order is observed at the cell center with even kth order polynomial approximations
2D non-isotropic diffusion equation
0)()( =++−++ yyxyxxyxt uuuuucu µ
have, wepartsby nintegratio , termmixed for the xyu
. dxdyvudxvuvdxdyuijiij I
yxI
jjx
Ixy ∫∫∫∫∫ −= +
−2/12/1|ˆ
2/1+jy .at ˆ 2/1±= jxx yuu
2D Buckley-Leverett equation
)()()( yyxxyxt uuugufu +=++ ε
−+−−
=
−+=
22
22
22
2
)1())1(51()(
)1()(
uuuuug
uuuuf
<+
=.otherwise ,0
,5.0 ,1)0,,(
22 yxyxu
Outline
Introduction of direct DG method as a diffusion solver
Derivation of DDG method Numerical flux coefficients Relation to interior penalty DG (IPDG) method DDG scheme for nonlinear diffusion DDG method on triangular meshes
Advantages of DDG method
Maximum-principle-satisfying or positivity-preserving Super convergence Elliptic interface problem
Current and future research work Deep Neural Network as an ODE solver
Maximum-principle-satisfying and positivity preserving 3rd order DDG methods
),()0,,( ,0))(()( 0 yxuyxuuuAuFut ==∇⋅∇−⋅∇+
On the discrete level we obtain the discrete maximum principle
. step timeand ),( allfor ,),( nnK tKyxMyxum Ω∈∈≤≤
2P
.12
1yx
t ,41
81 ,
23
010 βββ ≤
∆∆∆
≤≤≥
TheoremConsider the DDG schemes for the above convection diffusion equation s with quadratic polynomial approximations on unstructured triangular meshes. Given in the range of [m,M], we have solution average at next time level stay in the bound provided,
][][ˆ 10 xxxx uxux
uu ∆++∆
= ββ
𝑢𝑢𝐾𝐾(𝑥𝑥,𝑑𝑑) 𝑢𝑢𝐾𝐾𝑛𝑛(𝑥𝑥,𝑑𝑑)𝑢𝑢𝐾𝐾𝑛𝑛+1(𝑥𝑥,𝑑𝑑)
Z. Chen, H. Huang and J. Yan, JCP, 2016
Flow chart The flow chart for a high order DG scheme that satisfy the strict maximum:
∈∀≤≤
∆≤−
jnj
pnj
n
IxMxum
xCxutxu
,)(
|)(),(| Assume at step , we have
Step 1: evolve one time step and obtain
Step 2: apply limiter and obtain
jM, um 1nj ∀≤≤ +
nt
∆≤−
∀≤≤++
+
pnj
n
nj
xCxutxu
xMxum
|)(~),(|
,)(~
11
1
∫ ++
∆=
jI
nj
nj dxxu
xu )(1 11
difficult step!
easy step
2/1−jx 2/1+jxjx
- red curve:
- black line:
- green curve:
)(xunj
)(~ xu nj
nju
X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservationLaws, Journal of Computational Physics, 229(9):3091–3120, 2010.
?
Difficulty to obtain high order M-P-S for diffusion equation
For the convection part: only depends on solution values on the cell boundaries
0)( =+ xt ufu
)()( 2/12/11
−
∆∆
+= −
∧
+
∧+
jjnj
nj ufuf
xtuu ( ) )ˆ()ˆ( 2/12/1
1−+
+ −∆∆
+= jxjxnj
nj uu
xtuu
Hyperbolic Parabolic
0=− xxt uu
For the diffusion part: depends on solution derivative values on the cell boundaries
Y. Zhang, X. Zhang and C. Shu, M-P-S second order DG schemes for convection diffusions on triangular mesh, J. Comput. Phys., 2013.
∫∆=
jI
nj
j
nj dxxu
xu )(1
( ) ))(),(),(( )ˆ()ˆ( 112/12/11 xuxuxuHuu
xtuu n
jnj
njjxjx
nj
nj +−−++ =−
∆∆
+=
2/1−jx 2/1+jx
***
)(xunj
Bound solution average (algebraic method)
2/1+jx
***
2/3+jx
***
2/3−jx 2/1−jx
***
jI 1+jI1−jI
The Idea: pick as degrees of freedom and write out the quadratic polynomial in the Lagrange format.
)(xunj
−+
+− 2/12/1 ,, jjj uuu
),,,,,,,,(
),,,,,,,,( 2/312/12/12/12/112/31
↑↑↑↑↑↑↑↑↑=
= −++
++
−+
+−
−−−
+−
+
H
uuuuuuuuuHu jjjjjjjjjnj
MMMHummH nj =≤≤= + ),,( ),,(m 1
41
81
23
10 ≤≤≥ ββ and
][][)ˆ( 102/1 xxxjx uxux
uu ∆++∆
=± ββ
provided:
0=− xxt uu
2D Incompressible Navier-Stokes equations in vorticity stream-function formulation
( )
−==∆
∆=++
).,(, ,Re1)()(
xy
yxt
vuw
wvwuww
φφφ
Test problem: vortex patch problem.
We couple P2 continuous finite element for stream function and DDG method for vorticity.
Here is the vorticity of the velocity field ),( vuw
Taking the curl of the 2D incompressible N-S equations We obtain the vorticity stream-function formulation of the N-S equation.
)equations S-N(×∇
×∈+×∈−
==.elsewhere ,0
],4/7,4/5[]2/3,2/[),( ,1],4/3,4/[]2/3,2/[),( ,1
)0,,( ππππππππ
yxyx
tyxw
Background: chemotaxis and Keller-Segel equations
Motivation: study cell movement, aggregation and pattern formation phenomena
”chemotaxis”: …the influence of chemical substances in the environment to the movement of mobile species, for example bacteria…
Mathematical modeling: dates back to Patlak (1950’s) and Keller and Segel (1970’s)
Ω∂=⋅∇=⋅∇
+−∆=∆=∇+
on 0 )(div
nn cccc
c
t
t ρρ
ρχρρ
constanty sensitivit cchemotacti :ionconcentratctant chemoattra :),,(
density cell :),,(
χ
ρtyxctyx
Theoretical results: • Exists unique weak solution with smallness assumption on the initial• Exhibits point-wise blow-up pattern with certain initial• Blow-up at the boundary
Chemotaxis Keller-Segel system
+−==+ρρρρ
cccc
xxt
xxxxt )(
With polynomial approximation on chemical density, i.e, kP khh Vtxc ∈),(
xhc )(The spatial derivative only have kth order error under and norms.1L 2L
It is natural to introduce one extra variable, as LDG method, to approximate separately and plug it into first equation to obtain optimal (k+1)th order of accuracy.Available DG methods: Epshteyn and Kurganov (2008) (2009), Li, Shu and Yang (2016)
xcq =
It turns out DDGIC method can obtain optimal convergence without extra variable
Reason behind is our diffusion solver (DDG methods) have the hidden super-convergence property on its approximation to solution’s spatial derivative ),( tcx ⋅
xj+3/2xj-1/2 xj+1/2
jI 1+jI𝑐𝑐ℎ 𝑥𝑥 = 𝛽𝛽0
𝑐𝑐ℎ∆𝑥𝑥
+ 𝑐𝑐ℎ 𝑥𝑥 + 𝛽𝛽1∆𝑥𝑥 𝑐𝑐ℎ 𝑥𝑥𝑥𝑥
𝑐𝑐ℎ 𝑥𝑥
DDGIC scheme formulation for Keller-Segel system
−+==+
cccc
xxt
xxxxt
ρρρρ )(
−+++−=
=++−+−+
∫∫∫
∫∫
−+
+−+
−
−+
+−+
−
)()]([21)]([
21 |ˆ
0 )]([21)]([
21 )( |)ˆ)(ˆ(
2/12/12/12/1
2/12/12/12/1
dxwcwcwcdxwcwcdxwc
vvdxvcvfdxv
jjj
jj
Ijxjxx
Ix
jjx
It
jxjxxxI
xjjx
It
ρ
ρρρρρρρ
( )
∆++∆
=
−−+= −++−
][][ˆ
)()()(21)(ˆ
10 xxxx cxcx
cc
fff
ββ
ρραρρρ
DDGIC scheme formulation:
Numerical flux:
with
=
=+
+++
−++
−
ρρ
ρρ
2/12/1
2/12/1
)ˆ()(
)ˆ()(
jxj
jxj
cf
cf
xj+3/2xj-1/2 xj+1/2
jI 1+jI
defineduniquely is ˆxc
)1(21
1 +=
kkβ
))( denote( ρρ xcf =
𝛼𝛼 = max|( 𝑐𝑐𝑥𝑥)𝑗𝑗+1/2|
K-S system accuracy test
+−==+ρρρρ
cccc
xxt
xxxxt )(
xxc )( ρ ∫jI
xx dxvc ρ
W.-X. Cao, H. Liu and Z.-M. Zhang, Superconvergence of the direct discontinuous Galerkin Method for convection-diffusion Equations, Math. Numer. Methods Partial Differential Equations, 33(1): 290-317, 2017.
DDGIC method for Keller-Segel system
)(div
+−∆=∆=∇+
ρρχρρ
cccc
t
t
Ω∂=⋅∇=⋅∇ on 0nn cρ
420)0,,(
840)0,,()(42
)(84
22
22
=
=+−
+−
yx
yx
eyxc
eyxρ
The blow up time is measured at around . Blow up time measured by Epshteyn and Kurganov (2008) was .
3.0 × 10−41.21 × 10−4
super convergence on solution’s spatial derivative
Error norms:
=∈=−
B.C.Neumann zero ),cos()0,(],2,0[ ,0
xxcxcc xxt π
1)),(),()((max||)(|| )( LI
xxhjmMxxh vvdxtctcccj
∫ ⋅−⋅=−
mjxxxv )()( −=
(k+2)th and (k+3)th super convergence (Zhang and Yan (2017)) Main results:
])[()(][)( 10 xxhxh
hxh cxc
xcc ∆++∆
=∧
ββ
(k+2)th and (k+3)th super convergence with DDG method
P2 quadratic polynomials: analytical errors
1)),(),()((max||)(|| )( LI
xxhjmMxxh vvdxtctcccj
∫ ⋅−⋅=−
)1(21
1 +=
kkβ
W.-X. Cao, H. Liu and Z.-M. Zhang, Superconvergence of the direct discontinuous Galerkin Method for convection-diffusion Equations, Math. Numer. Methods Partial Differential Equations, 33(1): 290-317, 2017.
])[()(][)( 10 xxhxh
hxh cxc
xcc ∆++∆
=∧
ββ
(k+2)=4th and (k+3)=5th super convergence with P^2 approximations
Problem setup: elliptic Interface problem
Piecewise elliptic problem with solution jump and normal derivative jump conditions
−𝛻𝛻 𝛼𝛼 𝑥𝑥, 𝑑𝑑 𝛻𝛻𝑢𝑢± = 𝑓𝑓±, (𝑥𝑥, 𝑑𝑑) ∈ Ω±
𝑢𝑢 = 𝜙𝜙, 𝑜𝑜𝑛𝑛 Γ𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝜓𝜓, 𝑜𝑜𝑛𝑛 Γ
𝑢𝑢 = 𝑢𝑢0, 𝑜𝑜𝑛𝑛 𝜕𝜕Ω
𝑢𝑢 = 𝑢𝑢− − 𝑢𝑢+ = 𝜑𝜑 𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝛼𝛼−𝑢𝑢𝒏𝒏−− + 𝛼𝛼+𝑢𝑢𝒏𝒏+− = 𝜓𝜓across interface:
Ω−Ω+
𝝘𝝘
𝜶𝜶−
𝜶𝜶+ Ω− ∩ Ω+ = ΓInterface:
Computational domain: Ω = Ω− ∪ Ω+
𝛼𝛼 𝑥𝑥,𝑑𝑑 = 𝛼𝛼− 𝑥𝑥,𝑑𝑑 , (𝑥𝑥,𝑑𝑑) ∈ Ω−
𝛼𝛼+ 𝑥𝑥,𝑑𝑑 , (𝑥𝑥,𝑑𝑑) ∈ Ω+
𝒏𝒏−𝒏𝒏+Diffusion coefficient:
−Γ𝜓𝜓𝑣𝑣 𝑑𝑑𝑠𝑠 +
Ω−∪Ω+𝛼𝛼 𝑥𝑥, 𝑑𝑑 𝛻𝛻𝑢𝑢± 𝛻𝛻𝑣𝑣 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑 =
Ω−∪Ω+𝑓𝑓±𝑣𝑣 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑
Weak formulation with zero domain boundary condition:
Symmetric direct DG method for the elliptic interface problem with body-fitted mesh
−𝛻𝛻 𝛼𝛼 𝑥𝑥, 𝑑𝑑 𝛻𝛻𝑢𝑢± = 𝑓𝑓±, (𝑥𝑥, 𝑑𝑑) ∈ Ω±
𝑢𝑢 = 𝜙𝜙, 𝑜𝑜𝑛𝑛 Γ𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝜓𝜓, 𝑜𝑜𝑛𝑛 Γ
𝑢𝑢 = 𝑢𝑢0, 𝑜𝑜𝑛𝑛 𝜕𝜕Ω
A sample body-fitted mesh
Goal: Develop high order DG methodCurved triangular elements
𝑃𝑃𝑘𝑘 𝑝𝑝𝑜𝑜𝑝𝑝𝑑𝑑𝑛𝑛𝑜𝑜𝑝𝑝𝑝𝑝𝑎𝑎𝑝𝑝 𝑤𝑤𝑝𝑝𝑤𝑤ℎ 𝑘𝑘 = 2, 3 , 4
Symmetric DDG method for elliptic Interface problem
Symmetric DDG scheme formulation:
The Key is the definition of numerical flux on edges:(1) interior edges; (2) domain edges; (3) edges on the interface
𝒯𝒯ℎ = 𝒯𝒯ℎ− ∪ 𝒯𝒯ℎ+ ∪ 𝒯𝒯ℎΓWe first classify all triangles into three groups:
−𝛻𝛻 𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢± = 𝑓𝑓±, (𝑥𝑥,𝑑𝑑) ∈ Ω±
𝑢𝑢 = 𝜙𝜙, 𝑜𝑜𝑛𝑛 Γ𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝜓𝜓, 𝑜𝑜𝑛𝑛 Γ
𝑢𝑢 = 𝑢𝑢0, 𝑜𝑜𝑛𝑛 𝜕𝜕Ω
𝜕𝜕𝜕𝜕 ∈ ℰℎ = ℰℎ𝐼𝐼 ∪ ℰℎ𝐷𝐷 ∪ ℰℎΓ
ℰℎ𝐼𝐼 : 𝑠𝑠𝑠𝑠𝑤𝑤 𝑜𝑜𝑓𝑓 𝑝𝑝𝑛𝑛𝑤𝑤𝑠𝑠𝑖𝑖𝑝𝑝𝑜𝑜𝑖𝑖 𝑠𝑠𝑑𝑑𝑒𝑒𝑠𝑠𝑠𝑠 𝑝𝑝𝑛𝑛 Ω− 𝑎𝑎𝑛𝑛𝑑𝑑 Ω+
ℰℎ𝐷𝐷: 𝑠𝑠𝑠𝑠𝑤𝑤 𝑜𝑜𝑓𝑓 𝑑𝑑𝑜𝑜𝑝𝑝𝑝𝑝𝑎𝑎𝑛𝑛 𝑏𝑏𝑜𝑜𝑢𝑢𝑛𝑛𝑑𝑑𝑎𝑎𝑖𝑖𝑑𝑑 𝑠𝑠𝑑𝑑𝑒𝑒𝑠𝑠𝑠𝑠
ℰℎΓ: 𝑠𝑠𝑠𝑠𝑤𝑤 𝑜𝑜𝑓𝑓 𝑝𝑝𝑛𝑛𝑤𝑤𝑠𝑠𝑖𝑖𝑓𝑓𝑎𝑎𝑐𝑐𝑠𝑠 𝑠𝑠𝑑𝑑𝑒𝑒𝑠𝑠𝑠𝑠
Numerical fluxes definition on interface edges
Notice we have 𝒏𝒏− = −𝒏𝒏+, adding the flux contribution from two elements matchesthe interface flux jump condition in the weak sense.
𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝛼𝛼−𝑢𝑢𝒏𝒏−− + 𝛼𝛼+𝑢𝑢𝒏𝒏++ = 𝜓𝜓over the interface Γ:
𝛼𝛼𝑢𝑢𝐧𝐧 = 𝛼𝛼𝑢𝑢𝐧𝐧− , 𝑠𝑠 ∈ 𝜕𝜕𝜕𝜕−,𝛼𝛼𝑢𝑢𝐧𝐧+ , 𝑠𝑠 ∈ 𝜕𝜕𝜕𝜕+.we assign two separate values to the numerical flux 𝑠𝑠 = 𝜕𝜕𝜕𝜕− ∩ 𝜕𝜕𝜕𝜕+
𝑢𝑢 = 𝑢𝑢+ − 𝑢𝑢−
𝑢𝑢 = 𝑢𝑢− − 𝑢𝑢+
Ω−Ω+
𝝘𝝘
𝜶𝜶−
𝜶𝜶+
Numerical fluxes definition for element edges falling on the interface
Symmetric DDG scheme:
over the interface Γ solution jump condition: 𝑢𝑢 = 𝑢𝑢− − 𝑢𝑢+ = 𝜙𝜙
Continuity and coercivity of the bilinear form
Symmetric DDG global formulation:
Interface jump conditions are enforced weakly!
Circular interface with curved triangular elements
Accuracy test:optimal convergence orders with high order polynomial approximations
Three domains with two interfaces
Mixed interface jump conditions: 𝜙𝜙 ≠ 0 and 𝜓𝜓 ≠ 0, across Γ1𝜙𝜙 = 0 and 𝜓𝜓 = 0, across Γ2
High-contrast (multiscale) diffusion coefficients
𝛼𝛼−
𝛼𝛼+= 10−5 𝛼𝛼−
𝛼𝛼+= 105
−𝛻𝛻 𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢± = 𝑓𝑓±, (𝑥𝑥,𝑑𝑑) ∈ Ω±
𝑢𝑢 = 𝜙𝜙 = 0, 𝑜𝑜𝑛𝑛 Γ𝛼𝛼 𝑥𝑥,𝑑𝑑 𝛻𝛻𝑢𝑢 𝒏𝒏 = 𝜓𝜓 ≠ 0, 𝑜𝑜𝑛𝑛 Γ𝑢𝑢 = 𝑢𝑢0, 𝑜𝑜𝑛𝑛 𝜕𝜕Ω
High-contrast (multiscale) diffusion coefficientsThe interface is a circle and we have zero solution jump and none zero flux jump
The diffusion coefficient is two piecewise defined as:
The exact solution:
The focus of this example is to show symmetric DDG method obtains uniform optimal convergence orders that are independent of the inside/outside diffusion coefficient ratio
inside to outside ratio 𝛼𝛼−
𝛼𝛼+= 10−5, 10−4, 10−3,⋯ , 1,⋯ , 104, 105
Application: preserve the positivity of density and pressure profiles for Navier-Stokes equations
)2//(2/
0
)(
222
22
−+∂∂
=
++
∂∂
+
∂∂
uEuu
xPEuPu
u
xEu
tρκλ
λρρ
ρρ
)2/)(1( Pressure energy. totaland velocity density, are , , Here
2uEPEu
ργ
ρ
−−=
Now denote the conservative variables as , we rewrite N-S asTEm ), ,(ρ=w
0)()( =−+ xxxt DF www
Our goal is to prove 0)(,0)(given ;0)(,0)( nj
11nj ≥≥≥≥ ++ xPxxPx n
jnj ρρ
Or at least obtain the positivity of the polynomial solution average
. ,0)(,0)(given ;0)(,0)( nj
11nj SxxPxxPx ll
njl
nj ∈≥≥≥≥ ++ ρρ
Consider DDG discretization of N-S with polynomial solutions:Tn
jnj
nj
nj xExmxx ))(),( ,)(()( ρ=w
Conclusions Introduce the direct discontinuous Galerkin method
Show the advantages of DDG method Maximum-principle-satisfying or positivity-preserving Super convergence phenomena Elliptic interface problems
Current and future work DDG methods for strong nonlinear diffusion (compressible N-S
equations) Meshes being cut by the curved interface
Stokes interface problems
Thanks for your attention!