localized axial green’s function methods on free...
TRANSCRIPT
Do Wan Kim
Department of MathematicsInha University, Incheon, Republic of Korea
SRC, Yonsei Univ.(Dept. Math.), 7 Dec. 2015
Localized axial Green’s function methods on free grids
Question) Can we find the best approximation of solution of PDEs ?
Representation of solutions satisfying PDEs
Best approximation?
Green’s function in the same dimensional space
PDEs
Representation of solutions not satisfying PDEs
Locally approximating function
Something like FDM, FEMSomething like BEM, Spectral methods
not easy to find, limited in geometryeasy to find, not limited in geometry if
mesh or grid is well prepared
Best approximation of the solutions satisfying PDEs and
not limited in geometry without much additional effort
Green’s function in 1D, that’s enough
AGMAxial Green’s
function method
Localized AGM
Applicable to a large class of PDEs.
Easier Refinement, Adaptive, and Domain Decomposition, etc.
Arbitrary Cartesian grids are available even in complicated geometry.More available to convection-related problems
AGMs would be one of free grid methods.
Typical Discretization for
AGMs
Why (Localized) AGMs ?
O(h2)-convergence rate, though
1D
2D/3D
• Irregular grid• Arbitrary domain• No burden in
generating axial lines
• Innovative
Axial Green’s function method(AGM)
Irregular discretization with lines is available?
Axial lines
local axial lines parallel to axes
They can be extended up to boundary.ß This is the original AGM
a cross point
Local axial lines for AGMs
Y x
X y
General elliptic problems in 2D/3D (IJNME,2008) —- function coefficients
Stokes flows in 2D (JCP, 2011) —- a system of equations
Convection-dominated problems (JCP, 2014) —- hyperbolic phenomena
AGMs
Mass/heat transfers Navier-Stokes flows(High Reynolds — convection dominated case) Euler flows MHD …etc
Why the convection-diffusion equation? The convection and the diffusion are essential phenomena in nature, for instance,
Difficulties happen to the numerical methods for convection-dominated cases. In general, we need or encounter
up-winding scheme >> convergence rate and accuracy deteriorate. SUPG (streamlined-upwind-Petrov-Galerkin): FEM Theoretical convergence rate O(h^{3/2}) >> adaptive method can elevate the order. complicated geometry >> complex boundary layer. Irregular grid >> down the convergence rate.
What happens to the convection-dominated cases?
AGM seemingly doesn’t have to employ something like up-windings for stability because the axial
Green’s function has all it would need.
Localized AGM for Convection-Diffusion equation
Separating PDE into axial derivatives
ODEs on local axial lines (x, y)
Green’s function for adjoint differential operator
boundary values
Solution formula for ODE using 1D Green’s function
t� t+⌧
� (✏ut)t + a(t)ut = r(t)
1D Green’s function for general C-D equation
Solve these equations using 1D Green’s function
Representation of the solution by 1D Green’s function
Define 1D integral operators and boundary value transforms
A couple of 1D integral equations at each cross point
1D integral equations at each cross point
discretized at each local axial line parallel to axes
Discretizing 1D integral equations
1D piecewise linear approximations
If needed, for u(x) we do the same approximation.
1
Approximation of solutions
Algebraic equations on each cross point
Localization
This localization drastically reduce the computational cost evaluating 1D Green’s function.
Positive convection
Localized Green’s functions
Negative convection
AGM vs. FDM on uniform
Assume uniform axial lines(LAGM) vs. uniform grid(FDM).
Assume the following
Degenerate Case: elliptic problem
Quadrature effect in C-D Case
A-rule for quadrature is best.
Comparison to Up-wind scheme
Interior and boundary layers
Exact limiting solution
U = (1/2,p3/2)
✏ = 10�4
f = 0
irregular axial lines
40x40
80x80
Peanut-shaped Domain
U = (1, 1)
✏ = 10�4
(a) uniform 1580 cross points
(b) (h-2h) 1580 cross points
(c) random 1580 cross points
Peanut-shaped Domain
Complicated domain
Steady 2D Stokes flow
FEM FDM/FVM BEM
Mesh Unstructured Grid Boundary Mesh
LBB-satisfied Elements
Staggered Grid Fundamental solution
Saddle-structured matrix
Saddle-structured matrix
Full Matrix
AGM for the steady Stokes Flows
Formulation for Steady Stokes flow in 2D
Starting point in AGM for Stokes flow(possibly extended to 3D)
Introduce new variables
AGM formulation
Steady Stokes flow
Each PDE is seen as an ODE on each axial line.
1D Green’s function can be available.
Axial Green’s Functions
X-Axial Green’s Function
Exact Green’s Function
Y-Axial Green’s Function
Velocity representation w.r.t. 1D Green’s function (called Axial Green’s function in 2D/3D )
Representation formula of the flow velocity
Integration by parts for the pressure term
Divergence free condition Eq. (a)
Eq. (b)
Derived Integral Equations from momentum equations
Integral Equations from the continuous velocity
Divergence free condition
We need one more equation to complete the formulation.In order to do that, we calculate the direct differentiation for the
velocity.
Direct Differentiation of the velocity
The sum is valid at the cross point of axial lines and has to be 0 there
Explicit relationship ! New pressure correction schemes
AGM formulation for the Steady Stokes flow
x-momentum
y-momentum
Divergence free
Integral Transformations
Type –I derivatives
Type –II derivativesSomething like harmonic
Integral equations for Steady Stokes flow
Using the same scaling
It is a system of function equations on the interior of the fluid domain.
These formulations are not discrete but analytical.
The symbols imply integral transformations.
Matrix structure for Steady Stokes flow
How to discretize these 1D integral equations for the unknowns ?
Discretization
Linear basis for the approximations
Explicit Pressure Correction Algorithm: S-AGM(I)
is given
Pressure update
Projection Step
p⇤(n)
Implicit Pressure Correction Algorithm: S-AGM(II)
Pressure update
subject to
Projection Step
is givenp⇤(n)
Both algorithms are stable in the relaxation range . Moreover, the optimal value is near 1.7 ~ 1.9.
0 < � < 2
Lid-driven square cavity flow
Lid-driven square cavity flow
Lid-driven square cavity flow
Rotating flows(Circular cavity flow & Eccentric rotating cylinders)
Relaxation effect and convergence for circular cavity flow
Flow patterns( Uniform & Adaptive )
Uniform 4117 cross points
Adaptive 4784 cross points
Numerical Analytic
Relaxation effect and convergence for the eccentric rotating cylinder flow
Flow illustration for the eccentric rotating cylinder flow
3006 cross points
Matrix System of S-AGM
The # of nonzero elements
Size of matrices in S-AGM
Wave-bottomed cavity flow
JCP, (2006)by D.L. Young
Flow in a Complicated Domain
in-flow
out-flow
U
Pressure
V
Vorticity
out-flow
in-flow
0
5
5-50
Finer axial lines
Concentrated Axial lines on a region of interest
Adaptive or refinement
• Axial lines Independently constructed on two regions with interface, we can calculate the numerical solutions.
• At the node point on the interface, we can consider imaginarily extended axial lines that are needed.
Refinement
Matching axial lines
Assume .
Take .
Consider the cases .
2nd order convergence
Non-matching axial lines
Take .
Assume
Two regions of interest are separated with respect to
Consider the cases .
2nd order convergence
Interface Interface
u ru u ru
Interfacial problems?
By using the 1D Green’s function, the solution of the elliptic problem
Representation of the solution of 1D elliptic problem
for a finite set of singular points, can be represented as follows:
Or, equivalently
The error of the numerical solution of the following 1D elliptic problem
Error estimate of 1D elliptic problem
Theorem
is estimated as follows
If we use the exact instead of , then the source of error comes only from the numerical integration.
Numerical Example for 1D elliptic problem
Numerical Example for 1D elliptic problem
1-GP 2-GP
3-GP 4-GP
Schematic figure to apply 1D Green’s function to 2D/3D elliptic problems
Section along axis
Its boundary point
Difference acorss a point
Unit normal and unit tangent vectorsInterface
1D Green’s function theory for 2D elliptic problems
If using the 1D Green’s function for 2D(or 3D) elliptic problem, i.e.,
then the following integral representations hold, for and ,
1D Green’s function theory for 2D elliptic problems
For the 2D elliptic problem
If we define the function , then we have the following integral equation
Integral representation for Interface conditionfor 2D elliptic problems
Using the given interface condition for a smooth interface,
we have the following integral equation
where
1D Green’s function theory for 3D elliptic problems
CorollaryFor the 3D elliptic problem
If we define the functions
Then we have 3 integral representation formula as in the same manner in 2D case. Therefore, we can make 2 integral equations and 2 integral representations for interface condition.
Unknowns in 3D
Question) Steady Navier-Stokes flows
Question) Unsteady Navier-Stokes flows
Question) Maxwell Equations ?
Question) Boundary conditions ?
So many works to do in the future.
etc…
Question) Mathematical foundations ?
Thank you
We should do better
works than ever before.