Chapter 5. Basics of Higher-order Discretization Methods
Chap. 5-1. Comparison of Discretization Methods_Revisited
Discretization Methods Available
FDM (Finite Difference Method)
Direct discretization of the differential form by assuming point-wise values defined at each grid point
Simple, efficient and easy to discretize
One-dimensional (or dimension-by-dimension) interpolation and transformation to computational coordinate are essential. not suitable to complex geometry
FVM (Finite Volume Method)
Integration of the differential form of conservation laws over finite computational cell
Apply control volume analysis to each computational cell defined by (∆x, ∆t) using cell- averaged quantities.
, at with ( ),
( ) 0 j j j 0 plus a suitable time-discretization
jq j q
q f x x j q
h x t
dq c f q f q
t x dt h
1/2 1/2
integration discretized form using
over ( , ) cell-averaged quantities
1
1 2 1 2 1 1
( ) ( )
0 0
,..., ,...,
n j
n j
t t x
h t t x
n n n n
n n
j j p j q j j p j q
j j
q f q q f q
t x t x dxdt
F q q F q q
q q
t
1 2 1 2
1
1 2 1 2
0 or ( ,..., )
1 1
with , , ,
j n j n
j n n
j p j q
x n n t t
j j j
x t
dq L q q
h dt
q x t dx q f q x t dt F
x t
Chap. 5-1. Comparison of Discretization Methods_Revisited
FVM (Finite Volume Method) (cont’d)
Purely local analysis without assuming any grid structure or the shape of computational cell
suitable to complex geometry
Cell-wise interpolation for high-order accuracy not flexible on general unstructured grids
FEM (Finite Element Method)
Weak formulation (or weighted-residual formulation) from the differential form of conservation laws over each computational element
Local expansion of solution using basis functions followed by orthogonal projection of the residual (or error) to test function space
Local formulation without assuming any grid structure or the shape of computational cell suitable to complex geometry
Local expansion of solution with multiple DOFS flexible for higher-order approximation on general unstructured grids
Uncertainty on conservative and accurate treatment of the boundary integral
... 0
( ) ( )
1
( ) 0 0 on for , 1 ( )
approximate ( , ) on using basis functions, ( , ) ( ) ( ) evaluate temporal and
Tk i
k k k
dx i
i i k i
T T T
ndof n i
k k i
i
q f q d
q dx f dx f dx T i ndof n
t x t dx
q x t T q x t q t x
spatial integral terms by numerical quadrature
Chap. 5-1. Comparison of Discretization Methods_Revisited
Value of high-order accuracy
Ex) Behavior of computed solutions by changing order-of-polynomial approximation(N) mesh size(K)
0
1 1 1
1 2
2
0 with [0, 2 ], 2 , ( ) ( ,0) sin(2 )
- With periodic BC and a fixed time step for all cases, computed results shows i) h ( N ) ( ) N ( ) N with = targ
q q
a x a q x q x x
t x
q q O h C T h c c T h T
2
et time
ii) computational cost ( ) ( 1) with 2 , = order-of-approximation - Comparison of computed results reveals that higer-order approximation is beneficial for the cases requi
C T K N h N
K
ring i) highly accurate solutions, ii) long-time integrations
< Scaled computation cost >
N \ K 2 4 8 16 32 64
1 1.00 2.19 3.50 8.13 19.6 54.3
2 2.00 3.75 7.31 15.3 38.4 110.
4 4.88 8.94 20.0 45.0 115. 327.
8 15.1 32.0 68.3 163. 665. 1271.
16 57.8 121. 279. 664. 1958. 5256.
(~ O(h ) with n > 2)n
Strategy for High-Order Accuracy
One unknown (or one DOF) per one cell approach
Within a cell , approximate the exact solution as a polynomial by using Taylor expansion with neighboring computational cells (or grid points)
nth-order polynomial approximation within
FDM: direct Taylor expansion, compact difference
FVM: TVD-MUSCL, ENO/WENO, MLP reconstruction
Cell-interface values estimated from are used to evaluate a numerical flux.
+ Relatively simple, easy to understand, and suitable for coding
- Non-local computational stencil
As a result, when is higher than linear polynomial,
Hard to treat boundary conditions, hard to handle 3-D complex geometry, and hard to exploit parallel computations
Multiple DOFs per one cell approach
Within an element , approximate the exact solution as a polynomial by a linear combination of local shape functions (or basis functions).
Chap. 5-1. Comparison of Discretization Methods_Revisited
1 1 1 2 1 21
( , , , , ) , with
n l
k k l j a j a j b j b k k
l
q x q c q q q q x x x x a b n
Tk q x( ) q xk( )
Tk
k( ) q x
k( ) q x
Tk
Tk q x( ) q xkh( )
1 ( ) 1 2 1 2 1( ),
h n l
k k l k k
l
q x q
x x x x
Chap. 5-1. Comparison of Discretization Methods_Revisited
Multiple DOFs per one cell approach (cont’d)
nth-order polynomial approximation within
Each coefficient is regarded as an unknown or a degree of freedom (DOF).
Finite element discretization based on weighted residual formulation is commonly adopted to determine the evolution of .
+ Highly local construction compact stencil As a result,
Only nearest elements sharing a common cell interface/vertex are necessary → amenable to parallel computation
Easy to implement boundary conditions
Easier to tackle 3-D complex geometry
- More mathematical background and more efforts to implement it into a code
- More delicate numerical treatment for temporal and spatial integral terms
- More data storage and/or more computational costs in general
Overcome via computational techniques such as parallel computing or h/padaptation Tk
( )l
qk
( )l
qk
Tk
…
Discretization of 1-D SCL,
Weighted residual formulation
Approximate solution of within the cell
Choice of the polynomial space and basis function determines the nature of solution.
Approximate solution does not satisfy the governing equation exactly.
Minimize the residual, R,in the weighted integral sense over the test function space
Choice of the weight function determines the nature of the discretized equation.
Discrete Fourier spectral method:
Galerkin approximation:
Discontinuous Galerkin (DG) Method
Integration-by-parts over the cell with a compactly supported
a weak form:
Chap. 5-2. Basic Formulation
t x
q + f (q) = 0
( ) ik xˆj
w xj e ( ) ( )
i i
w x x
( ) ( )
1
( , ) kh( , ) ndof n kl ( ) ( ), where { }l l n and ( ) dim( )n 1
l
q x t q x t q t x V ndof n V n
1 1 ( )
( ) ( ) 0 over the whole domain elem for weighted functions { ( )}
k
N h
k i i i i i ndof n
T R q w x D T w x
( ) ( )
Non-zero residual: 0 ( )
h h h
k k k h
k
q f q q f q
t x t x R q
( ) ( ) 0 ( ) 0
ˆ 0
k k k k
k k k
i t x i t i x i
T T T T
i
i i
T T T
R q w x q f q dx q dx f dx
q dx f n dx f dx
t x
( , )
q x t Tk [xk1/2,xk1/2]
Vn l
( , )
h
q x tk
wi
T
k
i( ) x
DG approach to treat the boundary integral
Approximate solution allows to be discontinuous, in the sense of , at the cell interface .
→ f is not uniquely determined at the cell interface.
If , Thus, the weak form becomes or the finite volume discretization. → In order to maintain conservation at , f is
approximated by a conservative numerical flux H from finite volume method.
Spatial approximations
Chap. 5-2. Basic Formulation
i 1
- : the value of at the boundary obtained from ( ) of the cell
: the value of at the boundary obtained from ( ) of the adjacent ce
( , ) 0 with
k k k
k k
k
i
i i
T T T
q q T q x T
q q T q x
q dx H q q dx f dx
t x
ll sharing Tk
( ) ( )
1 ( ) ( )
1
( , ) ( ) 0
( ) ( , ) ( ) 0
( ) ( , )
k k k
k k k
k k
h h h h h h i
k i k k k i k k
T T T
ndof n
l h h h h h i
k l i k k k i k k
T T T
l ndof n
l h h h
k T l i T k k k i
l
q dx H q q dx f q dx
t x
q t dx H q q dx f q dx
t x
q t dx H q q dx
t
( )( )
1
( ) 0
Thus, we have ( , ) ( ) 0.
k
k k k
h h i
k k
T ndof n l
h h h h h
k i
i l k k k i k k
T T T
l
f q dx x
dx q H q q dx f q dx
t x
Tk
Tk
C0
( ) ( ) (1)
1
( , ) ndof n ( ) ( ) ( ).
h l
k k l k
l
q x t q t f x q t
dqk(1) f k 1/2 0dt
Semi-discrete form with a matrix-vector notation
Strong form by taking integration-by-parts once more
Weighted residual is dependent on the choice of numerical flux H and test function
Smoothness of is not essential.
Various approaches to remove the boundary integral term
Collocation penalty approach, using Dirac delta functions
Choice of Basis Function
Classification of polynomial basis functions
(Option 1) Modal basis function to represent a specific solution distribution (or mode shape)
Solution of a singular Sturm-Liouville problem
Chap. 5-2. Basic Formulation
( ) ( )
i x x yi
( )
( ) ( )
( ) ( )
( , ) ( ) ,
where , ( ) , and ( )
k k
k
h h h h h
k
k T k k k T k k
j
k T i j ndof n ndof n k k ndof n i ndof n
d H q q dx f q dx
dt x
dx q t x
q Φ
M Φ
M q Φ
0
or ( , )
k k k k k
k k
h h
h h
i k k
i i i k k i
T T T T T
h
h h h h
k k
k T T k k k k
q f
q dx H dx f dx dx f H dx
t x t x
d f
dx f H q q dx
dt x
M q Φ Φ
i
i
2 2
2
( ) ( )
(1 )d n x 2 d n x ( 1) ( ) 0 for n [ 1,1], 0
x x n n x x n
dx dx
Legendre polynomial( ) is a special case of the Jacobi polynomial( ) which is a solution of the general singular Sturm-Liouville problem.
Chap. 5-2. Basic Formulation
2
2 3 4 2
0 1 2 3 4
( ) 1 [( 1) ] (Rodrigues' formula) gives a sequence of -order Legendre ploynomials 2 !
1 1 1
as ( ) 1, ( ) , ( ) (3 1), ( ) (5 3 ), ( ) (35 30 3),...
2 2 8
Notable prop
n
n th
n n n
x d x n
n dx
x x x x x x x x x x x
1 1
1 1 ( 1)
erties
- Orthogonality: 2 and 0 with (1) 1, ( ) ( 1) ( ), 0
2 1
[ ] with and / || || becomes the identity matrix.
- Recurrence:
k
n
i j ij j n n n
k ij ij T i j
dx dx x x n
i
m m dx
M
1 1 1 1
1 -1
( 1) n ( ) (2 1) n( ) n ( ), d n ( 1) n d n , and d n d n (2 1) , n 1
n x n x x n x n x n n
dx dx dx dx
< Example of modal basis functions >
(Legendre polynomials)
< Example of nodal basis functions >
(Cubic Lagrange polynomials)
n Pn( , )
(Option2) Nodal basis function
Chap. 5-2. Basic Formulation
2 ( , ) ( , )
2 ( , )
2
( , )
2 ( , )
( ) ( )
(1 ) [ ( 2) ] ( 1) ( ) 0 for [ 1,1], 0,
and ( , ) 1 or (1 ) ( ) ( ) ( 1) ( ) ( ) 0 with ( ) (1 ) (1 )
n n
n
n
n
d P x dP x
x x n n P x x n
dx dx
d dP x
x w x n n w x P x w x x x
dx dx
P
( , ) 2
1 ( , ) ( , ) ( , )
1
( 1) 1
( ) [ ( )(1 ) ] (Rodrigues' formula) or a form of hypergeometric function 2 ! ( )
Notable properties
- Orthogonality: ( ) ( , , ) with ( ) ( 1)
n n
n
n n n
n
i j ij n n
x d w x x
n w x dx
P P w x dx f i P x P
( , )
( , ) ( , ) ( , ) ( , )
1 1 1
( , )
( 1, 1) 1
( ), 0
- Recurrence: ( ) ( ) ( ) ( ) and
( 1)
with ( , , ), ( , , ), 1
2 - Jacobi po
n n n n n n n
n
n n n n n
x n
xP x a P x b P x a P x
dP n
P a a n b b n n
dx
(0,0)
lynomials contain other orthogonal polynomials as a special case,
and they are quite useful for Gauss-like quadratures and construction of multi-dimensional basis.
. Pn ( )x n( ) (Legendre x
( 1/2, 1/2)
polynomials)
. Pn ( )x T xn( )con n( ) with xcos , - (Chebyshev polynomials)
( ) 1
Lagrange polynomial by interpolating some selective points - ( ) Thus, ( ) by definition.
- is not diagonalized.
- Typically employed in flux reconstruction app
ndof n
j
i i j ij
j i j
k
x x x x
x x M
roach such as FR/CPR, ESFR