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PinT2018: /2018/5/4 ROSCOFF Convergence Acceleration of the PinT Integration of Advection Equation using Accurate Phase Calculation Method Mikio Iizuka Kyushu University, RIIT Research Institute for Information Technology Kenji Ono
Mikio Iizuka
Kyushu University, RIIT（Research Institute for Information Technology）
Kenji Ono
Kyushu University, RIIT（Research Institute for Information Technology） RIKEN Advanced Institute for Computational Science, The University of Tokyo, Institute of Industrial Science, Japan
PinT2018: /2018/5/4
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Convergence Acceleration of the PinT Integration of Advection Equation using Accurate Phase Calculation Method
Acknowledgement
This work was supported in part by MEXT (Ministry of Education, Culture, Sports, Science and Technology ) as a priority issue (“Development of Innovative Design and Production Processes that Lead the Way for the Manufacturing Industry in the Near Future” ) to be tackled by using Post ‘K’ Computer. This research used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID:hp170238).
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Outline
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⚫Int ntro rodu duct ction
⚫How
do we we de deve velop
very a acc ccurate ph phase ca calcl clation
hod d ?
⚫ Method imp mproving t the c cal alculat ation me method of
ad advection ⚫ Check impacts of conventional methods improvement
⚫Parareal ca calcu culation
⚫Summary and nd Future W Wor
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Basis direction of our PinT research
research Engineering Viewpoint (CFD etc)
research
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Engineering Viewpoint (CFD etc)
Basis direction of our PinT research
Loves widely usable method, especially the method usable to hyperbolic PDEs
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Now, mainly, Advection eq.
Basis direction of our PinT research
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Love simple.→ Pararael method is better. do not like complex platform or frame work. Now, mainly Advection eq.
Basis direction of our PinT research
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Love simple.→ Pararael method is better. do not like complex platform or frame work.
Would you change your solver to incorporate your cord in PinT-Frame?
Basis direction of our PinT research
Now, mainly Advection eq.
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Love simple.→ Pararael method is better. do not like complex platform or frame work.
NO! NO! NO! NO! NO! NO!
Would you change your solver to incorporate your cord in PinT-Frame?
Basis direction of our PinT research
Now, mainly Advection eq.
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Loves simple.→ Pararael method is better. do not like complex platform or frame work.
Parareal
Basis direction of our PinT research
Now, mainly Advection eq.
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Loves simple.→ Pararael method is better. do not like complex platform or frame work. Have no interest in parallel method or HPC! → Have an interest in numerical method at least.
Parareal
Basis direction of our PinT research
Now, mainly Advection eq.
Engineering Viewpoint (CFD etc)
research
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Loves widely usable method, especially the method usable to hyperbolic PDEs Love simple.→ Pararael method is better. do not like complex platform or frame work. Have no interest in parallel method or HPC! → Have an interest in numerical method at least.
Parareal
Numerical method of advection
Basis direction of our PinT research
Now, mainly Advection eq.
Engineering Viewpoint (CFD etc)
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DE ODE Parabolic PDE Hyperbolic PDE
Parareal convergence
Relatively good BAD
Issue of the parareal method for hyperbolic PDEs:
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⚫ Hyp yperbolic PDEs r represents ts wav ave p phenomena. a. ⚫ If th there i is Phas ase D Difference between fine and coarse solver’s result t → Os Oscillat ations ap appear ars at at th the e edge of ti time s slice. ⚫ That at g give ves dam amag age th the conve vergence of th the par arar areal al meth thod.
*M. Gander and M. Petcu, Analysis of a Krylov subspace enhanced parareal algorithm for linear problems, ESAIM Proc, 25, (2008), 114-129.
Reducing the phase difference between fine/coarse solvers
Our challenge Why so bad?
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Example of oscillations in parareral iteration for advection equation
Profile of Φ at t=0.5 after 10 iteration
with time-coarsening ratio Rfc=25 without relaxation of iteration
Oscillation amplitudes much depend on numerical integration methods (may be accuracy of pahse calcultion).
0.5 1 1.5 2
200 400
0.5 1 1.5 2
5
Φ
x x
Fine/coarse solver :TVD/CN Fine/coarse solver CIP3rd method
Φ
x
step step C=1
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We e stu tudi died ed t the he impa pact ct of
phas ase e di differ eren ence ce on
the e pa parareal co conv nvergenc nce using ng m mos
ple p prob
This gives the exact phase to fine/coarse solver.
This results shows that very very small phase difference causes the convergence difficulty.
(a) Most simple problem:
→ Simple harmonic motion → Simplest hyperbolic PDE
(b) Time integrator:
Modified Newmark-β Method This method can give the exact phase for the simple harmonic motion independent on time step width by the modified δt’, δT’. We tried to check the effect of phase difference by adding the erro to coarse solver by value ε. Fine solver Coarse solver
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We e stu tudi died ed t the he impa pact ct of
phas ase e di differ eren ence ce on
the e pa parareal co conv nvergenc nce using ng m mos
ple p prob
This gives the exact phase to fine/coarse solver.
This results shows that very very small phase difference causes the convergence difficulty.
(a) Most simple problem:
→ Simple harmonic motion → Simplest hyperbolic PDE
(b) Time integrator:
Modified Newmark-β Method This method can give the exact phase for the simple harmonic motion independent on time step width by the modified δt’, δT’. We tried to check the effect of phase difference by adding the erro to coarse solver by value ε. Fine solver Coarse solver
Therefore, we focus on development of
th the me meth thod th that at reduc uce ph phas ase difference ce betw tween fine/co coar arse se so solv lver by ap appl ply th the very ac accu curat ate ph phas ase ca calc lcla lati tion me meth thod to to fine/co coar arse se so solv lver.
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This research approach:1
Approach based
parareal method
⚫ δT δt ⚫ Reduce the time span: T → Σ^{nc}_{l=1} T_{l} ⚫ etc
Approach based on the engineering method Dramatically Improving the conventional calculation method of advection equation to
increase the phase accuracy
Speedup Residual Conventional method Ｉmprovement method
There is a limit in the us
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Methods overview of advection term calculation
Conventional main method
1st: CIP scheme improvement: advecting the much phase information by the gradient and curveutre of value Φ.
Stabilization: numerical damping Accuracy: Space and time higher order terms Advecting only amplitude of variables
2nd: STRS scheme: achieving the stabilization and error elimination using “space and time reversal symmetry “ base on the physics. 3rd: Hybrid of CIP method and STRS scheme
Methods that are tried in this study
Main issue : Gap of phase accuracy between fine and coarse solver
Hybrid: STRS-CIP
not yet success There is a limit in the use. not enough 8
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This research approach:2
Grid based wave number : k=2π/λ= 2π/m/Δx
Conventional methods of advection equation lose phase accuracy for high grid based wave number waves except CIP3rd method.
Dispersion relations numerical calculation for advection equation.
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Simple and Typical Benchmark Problem including high grid based wave number waves
Curved Step 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1
x
Φ(x)
Step Curved
Most tough problem: Including broad and high grid based wave number waves Most simple problem: Including high grid based wave number
Analytical CIP3rd CIP5th
1.8 1.85 1.9 1.95 2
0.2 0.4 0.6 0.8 1
Step wise advection problem
Sin wave advection problem with very rough grids
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Simple and Typical Benchmark Problem including high grid based wave number waves
Curved Step 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1
x
Φ(x)
Step Curved
Most tough problem: Including broad and high grid based wave number waves Most simple problem Including high grid based wave number
Analytical CIP3rd CIP5th
1.8 1.85 1.9 1.95 2
0.2 0.4 0.6 0.8 1
Step wise advection problem
Sin wave advection problem with very rough grids
Engineering peoples using CFD every time ask me that the parareal method work well for step wise shape or rough grid sin wave advection?
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Conventional calculation method of advection term
PPM : Piecewise-Parabolic Method ENO : Essentially Non-oscillatory WENO: weighted ENO
Non linear type Linear type( CFL-free form used by the Semi-Lagrangian scheme)
(A) Groupe using only variables amplitude (B) Groupe using variables and those gradients
CIP3rd method:
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◼ CIP method advects variable’s gradients as the phase information. ◼ The phase accuracy of CIP method is higher than other conventional methods for especially high wave number.
What is CIP scheme?
◼ . Constrained Interpolation Profile scheme?
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Up stream calculation and time integration
pefrormed by back-trace and shift operation (CFL free
formula is used here: we can easily use large δT for case solver)
Back-trace points finding
Back-trac ace： Shit o
ration
＊Considering the equation on the grid i
id = grid that is near i grid of cell (id, id-1) Upstream finding
30 ⚫ Space discretization： by the cubic interpolation function CIP 3rd method ⚫ Up-date(time integration) : by Semi-Lagrange scheme CIP 5th method (We developed it as more accurate CIP at this time.)
Detail of Formula
(gradient)
(gradient, curvature)
Ｐｏｉｎｔ by the ５ｔｈ interpolation function
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do j=1,Ny; do i=1,Nx do j=1,Ny; do i=1,Nx
cx = cx =0.5d0* 0.5d0*(v (v1(i 1(i-1,j, 1,j,1)+v1( 1)+v1(i,j,1) i,j,1)) ida = ida =i-int( int(cx cx*dt/dx) *dt/dx) xgi = xgi =-cx*dt cx*dt+d +dx*real(i x*real(i-ida) ida) ais = ais =sign(1 sign(1.0 .0,cx) ,cx) idam= idam=ida ida-in int( t(ais) ais) fv = fv = v1 v1(ida,j (ida,j,3) ,3) gfi = gfi =gf (id gf (ida, a,j,1) ! j,1) ! dfai/d dfai/dx ggfi= ggfi=ggf(id ggf(ida, a,j,1) ! d j,1) ! ddfai/d dfai/dx/dx x/dx dfi = dfi =v1 (id v1 (idam am,j,3) ,j,3)-v1 v1(ida,j (ida,j,3) ,3) aid1 aid1=-dx5*a dx5*ais is*( 6.0* *( 6.0* dfi & dfi & & & + 3.0* + 3.0*( g ( gf (ida f (idam, m,j,1)+ j,1)+ gfi) gfi)*dx*ai *dx*ais s & & & & + 0.5* + 0.5*( g ( ggf(ida gf(idam, m,j,1) j,1)- ggfi) ggfi)*ddx *ddx ) bid1 bid1= dx dx4* 4* ( (-15. 15.0* 0* df dfi & i & & &
(7.0*gf (ida f (idam, m,j,1)+8.0 j,1)+8.0* gfi) * gfi)*dx*ai *dx*ais s & & & &
( ggf(ida gf(idam, m,j,1) j,1)-1.5 1.5*ggfi) *ggfi)*ddx *ddx ) cid1 cid1=-dx3*a dx3*ais is*( 10.0* *( 10.0* dfi & dfi & & & + 4.0* + 4.0*( g ( gf (ida f (idam, m,j,1)+1.5 j,1)+1.5* gfi) * gfi)*dx*ai *dx*ais s & & & & + 0.5* + 0.5*( g ( ggf(ida gf(idam, m,j,1) j,1)-3.0 3.0*ggfi) *ggfi)*ddx *ddx ) v2 ( v2 (i,j,3) i,j,3)= = & &(((( (((( aid aid1* 1*xgi+ xgi+ bid1) bid1)*xgi+ *xgi+ cid1)*x cid1)*xgi+0.5 gi+0.5*ggfi) *ggfi)*x *xgi+gfi)* gi+gfi)*xgi+fv xgi+fv gfn ( gfn (i,j,1) i,j,1)= = & & ((( 5 ((( 5.0*aid .0*aid1* 1*xgi+ 4.0 xgi+ 4.0*bid1) *bid1)*xgi+3 *xgi+3.0 .0*cid1)*x *cid1)*xgi+ gi+ ggfi) ggfi)*x *xgi+gfi gi+gfi ggfn( ggfn(i,j,1) i,j,1)= = & & ((20 ((20.0*aid .0*aid1* 1*xgi+12.0 xgi+12.0*bid1) *bid1)*xgi+6 *xgi+6.0 .0*cid1)*x *cid1)*xgi+ gi+ ggfi ggfi
end do; end do end do; end do
Set of the advection parameters Calculation of the coefficient
function Update of variables
Code of CIP-5th method CIP-5th method is very simple
and we can easily develop CIP5th code based on CIP3rd method code.
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*Katsuhiro Watanabe, a novel framework to construct amplitude preserving wave propagation schemes, Japan Society for Industrial and Applied Mathematics Annual meeting (2010), 123-124.(written in Japanese)
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Space and Time Reversal Symmetry guarantees the CONSERVATIVENESS of the amplitude Φ. STRS scheme is based on a symmetry of advection equation. That symmetry is the PARITY CONSERVATIVENESS, which is expressed by following formula in CONTINUUM SPACE.
What is STRS(Space-Time Reversal Symmetry) scheme?
parity transformation (also called parity inversion) is the flip in the sign of coordinate パリティ変換 (parity transformation) は一つの座標の符号を反転させることである。 パリティ反転 (parity inversion) とも呼ぶ。
Parity Transformation CONSERVATIVE
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General formula:
How to construct the STRS scheme of linear type of advection differencing schemes
CONSERVATIVE Parity Transformation in the discretization space
We can convert it to STRS scheme mechanically.
This scheme gives (a) stable, (b) no damping of amplitudes numerical methods.
(A) (B.1) (B.2) ・1st step: Perform the Parity Transformation on RHS of eq.(A) ・2nd step: Replace LHS of eq.(A) by that. Then we get formula (B.1). Let’s check the STRS of eq.(B.1)
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Most simple example: STRS scheme of Upwind 1st order Stability analysis using fourier transform
Time development formula of value Φ for mode l in complex plane No damping of amplitude
36 We can adjust the numerical phase speed to correct phase speed for one mode. Numerical phase speed from physical dispersion relation
Kawamura and Kuwahara-3rd, central-4th etc. schemes can be transformed to STRS scheme as same way!
Solve and use it!
In this case, phase correction can be done as here.
However, phase adjustment is available for upwind 1st and one mode case, very special case only.
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STRS-CIP formula
The formula is Parity CONSERVATIVE, but this still dose not work. Reason why, not yet clear.
We can easily get STRS-CIP formula from CIP3rd scheme by the Parity Transformation.
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1.9 1.91 1.92 1.93 1.94 1.95
0.05 0.1
Then, we tried an approximation version : STRS-CIP3rd_mod.
◼ Method : 1st step: get the gradient g by CIP3rd. 2nd step: get the value Φ using STRS-CIP3rd formula with given gradient g. ◼ Check the improvement : ・ sin wave (5grids/wave) advection ・ Space [0,2]×Time: [0,2] ・ CFL = 0.1 x Φ CIP3rd STRS-CIP3rd_mod Exact STRS-CIP3rd_mod CIP3rd
Results of Φ distribution （t=2: after 20 cycles）
We can improve CIP3rdd by ＳＴＲＳ ａｐｐｒｏｘｉｍａｔｉｏｎ． But that improvement is small. Then, we skipped this one this study.
→ Future challenge.
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Benchmark: Step Shape Advection
41 ⚫ 1D advection of step shape ・speed c=1.0 ⚫ Space Space [0, [0,3]×Time: Time: [0, [0,0.5 0.5 or
2.25]
Numerical analysis condition
⚫ Num. of meshes: 300 → dx=0.01 ⚫ Width of time step →dt=0.005,0.0025,0.00125 → CFL=0.5, 0.25, 0.125 ⚫ Boundary cond Boundary condition ition：continuous continuous ⚫ Initial condition → x=0--0.5:Φ=1.0, x > 0.5: Φ=0.0
Advection numerical method Physical condition
Parameters of the test
⚫ CIP3rd vs vs C CIP5th th meth thod
Ch Check the CI CIP5th me method d p perfo forman ance
by CIP IP3rd vs s CIP IP5th method
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x Φ(x)
（a）Results of Φ distribution（t=0.5）
Φ(x) x
0.75 L=3.0 1.0
CIP5thCFL0.5 CIP5thCFL0.25 CIP5thCFL0.125 CIP3rdCFL0.125 CIP3rdCFL0.5
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06
0.2 0.4 0.6 0.8 1 1.2
Initial condition
CIP5th.CFL0.5 CIP5th.CFL0.125 CIP3rd.CFL0.5 COP3rd.CFL0.125 2.6 2.65 2.7 2.75 2.8 2.85 2.9 0.2 0.4 0.6 0.8 1
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CIP5th.T=2.cfd0.5.step.txt CIP5th.T=2.cfd0.125.step.txt CIP3rd.T=2.0.5.step.txt CIP3rd.T=2.125.step.txt 2.746 2.748 2.75 2.752 2.754 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
x Φ(x) x
CIP-5th method → Step shape is sharp. → Phase accuracy is better. ➔ Improvement has been achieved!
（*） Zoomed part
CIP5th.CFL0.125 CIP3rd: CFL0.125
Analytical
ZOOM
（b） Results of Φ distribution（t=2.25）
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45 ⚫ Advection of sin wave (one mode wave) ・Φ(x)=sin(2πmΔx((i-1)/λ+0.5))、λ=0.1 → g(x)= dΦ(x)/dx =2πm/λ cos(2πmΔx((i-1)/λ+0.5)) ・10grids/wave(m=10) or 5grid/wave(m=20) ・velocity c=1.0 ⚫ Space Space [0, [0,2]×Time: Time: [0, [0, 2]
Analysis condition: space and time descritaization
⚫ dx=0.01 or 0. dx=0.01 or 0.02 02、200 or 100 meshes 200 or 100 meshes →L=dx L=dx×200=2 200=2 ⚫ dt =0.001 or dt =0.001 or 0.002( 0.002(CFL=0.1 CFL=0.1) ⚫ Boundary cond Boundary condition ition：cyclic cyclic
STRS scheme vs Conventional scheme
⚫ TVD 3 TVD 3rd
rd （3rd rd order)
⚫ CIP scheme 3rd
rd order
⚫ CIP 5 CIP 5th
th (5
(5th
th
Test problem
Parameters of the test
No-damping and no phase error STRS scheme using phase adjustment for one mode
⚫ STRS STRS-Upwind 1st
st order
with phase adjustment (Exact for one mode wave)
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Results after T=2.0 (20 cycles )
Exact STRS phase adjustment Kawamura and Kuwahara 3rd TVD3rd CIP3rd CIP5th 1.85 1.9 1.95 2
0.2 0.4 0.6 0.8 1 Exact STRS phase adjustment TVD3rd CIP3rd CIP5th 1.8 1.85 1.9 1.95 2
0.2 0.4 0.6 0.8 1
10 grids/wave 5 grids/wave
Phase improvement has been achieved by CIP5th and STRS phase adjust cases
Φ x x
TVD 3rd after one cycle
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・δt: time step width of fine solver： set by the CFL condition: Δx/v > δt ・δT: time step width of coarse solver： δT >> δt
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Purpose of benchmark test
◼ Set the same method of advection calculation in fine/coarse solver ◼ Phase accuracy increases along TVD3rd → CIP3rd→ CIP5th→ STRS. ➔ phas ase difference d decreas ase!
Phase accuracy Corse solver with δT Fine solver with δt
Study the impact of the ph phase di differenc nce betwe ween n fine ne/co coarse sol
ver to
TVD3rd → CIP3rd→ CIP5th→ STRS with phase adjustment.
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Parareal_CN-TVD as reference.
Time loop
Numerical flux construction Time integrator: Crank Nicolson
Fine solver
Time loop
Numerical flux construction Time integrator: Crank Nicolson
Coarse solver
δt δT
Time loop
CIP function construction Time integrator: Semi-Lagrange
Time loop
CIP function construction Time integrator: Semi-Lagrange
Parareal codes for each methods
Parareal_CIP3rd, CIP5th Parareal_STRS
Time loop
STRS coefficient construction Time integrating: STRS-Euler-1st
Time loop
STRS coefficient construction Time integrating: STRS-Euler-1st
with phase adjustment same method in fine/coarse solver → δt << δT : only difference
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f(x)=0.5(1-tanh((x-xo)/xi)、 xi : width of step → xo=1.0, → xi =SQRT(2D/k)=SQRT(2/k): 0.035(k=1600),0.07(k=400)
Numerical test: Parameters
Test problem
⚫ (b) advection of step like wave ⚫ C = 1.0 and Space [0,3]×Time: [0, 2.0]
Space and time descritaization
⚫ dx=0.01, 200meshes (10grids/wave) →L=dx×200=2 ⚫ δt =0.001(CFL=0.1) ⚫ Boundary condition：continuous
PinT condition
⚫ Number of time slices: 20 ⚫ Time coarsening factor Rfc = 25 (δT= 0.025)
“Step” and “Smooth curves” are used as initial condition. → When smoothness of curve increases, the number of grid based high wave number waves decrease.
Initial condition ⚫ (a) advection of step shape
Curved Step 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1
x
Φ(x) See the initial condition bellow.
CN-TVD CIP3rd CIP5th 5 10 15 20 10-15 10-10 10-5 100 105
Iteration number Kpar
Residual res^ (Kpar)
Iteration number Kpar
CN-TVD CIP3rd CIP5th 5 10 15 20 10-15 10-10 10-5 100 105
Relaxation parameter α=1.0 52
Results : Residual during the parareal iteration：
＊CIP methods and reduce of the grid based high wave number waves improves the convergence. ＊CIP5th has not so much effectiveness than CIP3rd. → Reason why not yet clear ?
STEP SMOOTH CURVE
CN-TVD CIP3rd CIP5th 5 10 15 20 10-15 10-10 10-5 100 105
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Influence of parareal iteration realxation
Iteration number Kpar Residual res^ (Kpar) Relaxation: NO α=1
Relaxation is effective for residual rebound, but Not so much effective ?
CN-TVD CIP3rd CIP5th
5 10 15 20 10-15 10-10 10-5 100 105
Iteration number Kpar Relaxation: α=0.2
CN-TVD CIP3rd CIP5th 5 10 15 20 10-15 10-10 10-5 100 105 CN-TVD CIP3rd CIP5th 5 10 15 20 10-15 10-10 10-5 100 105
Step
Residual res^ (Kpar) SMOOTH CURVE
Residua rebound
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Φ(x)
Kpar 1 3 5 10 20 2.65 2.7 2.75 2.8 2.85 0.2 0.4 0.6 0.8 1 1 3 5 10 20 Kpar 2.65 2.7 2.75 2.8 2.85 0.2 0.4 0.6 0.8 1
CN-TVD Residual res^ (Kpar) Iteration number Kpar
CIP-5th looks not so much effective than CIP-3rd, really ? Then, check the profile of variable along iteration・・・
X X X
CIP-5th CIP-3rd Change of the profile Φ along Kpar. CIP-5th is very accurate even for Kpar=1.
Profile show that CIP-5th is effective even for first sate of the iteration! 28
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Test problem
⚫ Space [0,2]×Time: [0,2.0]
Analysis condition: space and time descritaization
⚫ dx=0.01、200meshes（10grids/wave） →L=dx×200=2 ⚫ δt =0.001(CFL=0.1) ⚫ Boundary condition：cyclic
PinT condition
⚫ Number of time slices: 20 ⚫ Time coarsening factor: Rfc = δt/δT = 6, 12, 24 (δT=0.006, 0.012, 0.024) ⚫ Advection of sin wave (one mode wave) ・Φ(x)=sin(2πmΔx((i-1)+0.5)) (m=10) → g(x)= dΦ(x)/dx=2πm cos(2πmΔx((i-1)+0.5)) ・Velocity c=1.0
Parameters of numerical test
TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010
57 Relaxation α=1.0
Results
TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010 TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010
Residual res^ (Kpar) Iteration number Kpar Iteration number Kpar Rfc=6 Rfc=12 Rfc=24
At the stage of iteration start, → Residual corresponding to the phase difference → Small phase difference gives small residual.
Iteration number Kpar
TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010
58 Relaxation α=1.0
TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010 TVD/CN CIP3rd CIP5th STRS 5 10 15 20 10-15 10-10 10-5 100 105 1010
Residual res^ (Kpar) Iteration number Kpar Iteration number Kpar Rfc=6 Rfc=12 Rfc=24
Along the iteration, → Smaller phase difference causes larger residual rebound!
Iteration number Kpar
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Good: we achieved very small residual at the
start stage of iteration for very tough problem.
Bad: along the iteration, smaller phase difference causes the residual rebound, reason why not yet unclear ???
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◼ We have achieved BIG STEP in the CFD method view. ◼ But, that BIG STEP dose not work well for the parareal
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◼ Now, I have tool that help us to study the impact of the phase difference to parareal convergence. Using that tool, we continue to develop the method for PinT
◼ Also, development of STRS-CIP scheme is challenge. Maybe, it gives another BIG STEP.
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