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ADER schemes for linear advection-diffusionproblems with nonlinear source term using
several flux reconstructions.
R. Castedo, A. HidalgoDpto. de Matemática Aplicada y Métodos Informáticos,
E.T.S.I. Minas, Univ. Politécnica de [email protected], [email protected]
Hyperbolic systems with source term – Castro Urdiales ´09 07/09/2009
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 2
AIM.
To compare the behaviour of ADER scheme with Cauchy- Kowaleski procedure (E.F. Toro & R.C. Millington, 2001) and ADER with local space-time discontinuous Galerkin (M. Dumbser, C. Enaux & E.F. Toro, 2008), when solving problems with stiff source terms.
07/09/2009
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 3
Contents.
1. The numerical scheme.
2. ADER approach.2.1. ADER-FV.2.2. ADER-DG.
3. Numerical results.3.1. Linear advection problem.3.2. Linear advection-reaction problem.3.3. The scalar model problem of LeVeque and Yee.3.4. Advective-reactive-diffusive linear equation: convergence and practical application.
4. Conclusions and future research.
07/09/2009
4
1. The numerical scheme.
∂∂t
U(x,t) +∂∂x
F(U(x,t)) = S(x,t,U(x,t))
Finite Volume scheme for hyperbolic systems in non-conservative form:
[ ]
1/2
1/2
1
11/2
1/2
11/2 1/2 1/2 1/2
1 ( , ) ,
1 ( ( , )) ,
1 1 ( ( , )) .
i
i
n
n
ni
ni
xn ni x
in tn n n n
i i i i i i in ti
t x
i in t xi
x t dxx
t t x t dtx t
x t dxdtx t
+
−
+
++
−
++ − + +
⎧ =⎪ Δ⎪Δ ⎪
= − − + Δ =⎨Δ Δ⎪⎪ =⎪ Δ Δ⎩
∫
∫
∫ ∫
U U
U U F F S F F U
S S U
along with an initial and boundary conditions. Integrating this equation in the following space-time control volume
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
Qi = xi−1/ 2 ,xi+1/ 2⎡⎣ ⎤⎦ × t n ,t n+1⎡⎣ ⎤⎦.
5
Fci+1/ 2 =1Δt
v(xi+1/ 2 ,τ )U (xi+1/ 2 ,τ )dτtn
tn+1
∫
Fdi+1/ 2 = −1Δt
K(xi+1/ 2 ,τ )∂∂x
U (xi+1/ 2 ,τ ) dτtn
tn+1
∫
⎫
⎬⎪⎪
⎭⎪⎪
Fi+1/ 2 = Fci+1/ 2 + Fdi+1/ 2
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
2. ADER approach.2.1. ADER-FV. 2.3. ADER-DG.
ADER-FV of E.F. Toro and V.A. Titarev, the time evolution is achieved via a Taylor series expansion (solution of the DRP in each cell interface) where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time (Cauchy-Kovalewski procedure) Not able to handle stiff source term.
Si =1Δxi
1Δt n S(U i (x,t)) dx dt
xi−1/2
xi+1/2∫tn
tn+1
∫ .
Source term expression inside each volume:
We discretize the space integral with N-point Gaussian rule, then we reconstruct values in each Gaussian point in analogous procedure to the flux evaluation (WENO) and finally for each point we perform the C-K procedure and replace time derivatives by space derivatives.
6R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
2. ADER approach.2.1. ADER-FV. 2.2. ADER-DG.
ADER-DG of M. Dumbser et. al., the time evolution is obtained via a local space-time Discontinuous Galerkin (DG) finite element scheme Able to handle stiff source term.
In order to describe the ADER-DG approach we will consider the simple scalar model equation:
f (u) = vu and S(u) = −qu, v > 0,q > 0,
Basis functions are defined by the space-time tensor product of the Legendre polynomials
Φk = Φk (ξ,τ ) = Ψ i (ξ) ⋅Ψ i (τ ).kΦ ( ), ( ).i iξ τΨ Ψ
∂∂τ
u + v *∂∂ξ
u = −q *u, with v* = Δtξx ⋅ v and q* = Δt ⋅q.
1
ˆ ˆ( , ) ( , ) : ( , )dN
i ii i l l l l
lu u u uξ τ ξ τ ξ τ
=
= = Φ ⋅ = Φ ⋅∑The local numerical solution inside each space-time control volumeiu :iQ
Modified governing PDE
Multiplication of the new PDE with basis functions, and integration over the reference element:
[ ] [ ]1 0
, , , * , * , .k i k i k i k i k iu w u v u q uτ τ τ ξ
∂ ∂Φ − Φ − Φ + Φ = − Φ
∂ ∂
7
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
∂∂t
u x,t( )+ ∂∂x
u x,t( )= 0, x∈ (-1,1), t > 0,
u(x,0) =
exp(−ln2(x +0.7)2 /0.0009), -0.8≤ x ≤ −0.6,1, -0.4 ≤ x ≤ −0.2,1− 10x −1 , 0.0≤ x ≤ 0.2,
(1−100(x −0.5)2)1/2, 0.4 ≤ x ≤ 0.6,0, otherwise.
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
and periodic boundary conditions.
Problem presented by E.F. Toro and V.A. Titarev in 2005.
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
WAF-SUPERBEE
RUSANOV
G-FORCE
8R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
Problem presented by V.A. Titarev in 2005.
The exact solution is:
9
∂∂t
u x,t( )+ ∂∂x
u x,t( )= 5u x,t( ), x ∈ (-1,1), t > 0,
u(x,0) = sin4 π x( ),
⎫
⎬⎪
⎭⎪
and periodic boundary conditions. u(x,t) = sin4 π (x − t)( )e5t .
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
Problem presented by LeVeque and Yee in 1990, is a scalar linear advection problem, with a nonlinear reaction term, which can be stiff.
The initial condition is:
10
∂∂t
u +∂∂x
u = −qu(u −1) u −12
⎛⎝⎜
⎞⎠⎟
, x ∈[0;1], t > 0,
with transmissive boundary conditions.
1 if 0.3,( , )
0 if 0.3.x
u x tx≤⎧
= ⎨ >⎩
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
With this initial condition, the source term is zero, and the analytical solution of the problem is:
u(x,t) = u(x − t,0).
11R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
x
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Exact sol.ADER3-DGADER4-DGADER3-FVADER4-FV
q = 10
x
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Exact sol.ADER3-DGADER4-DGADER3-FVADER4-FV
q = 100
Parameters:Courant = 0.75; Time = 0.3;
Cells = 100; Values of q: q = 10 and q = 100.
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
12R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
x
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Exact sol.ADER3-DGADER4-DGADER3-FVADER4-FV
q = 1000
Parameters:Courant = 0.75.
Time = 0.3.
Cells = 500; Values of q: q = 1000.
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.
3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
13
∂∂t
u x,t( )+1.5∂∂x
u x,t( )− 0.8∂2
∂x2 u x,t( )= 3u x,t( ), -10 < x < -10, t > 0,
u(x,0) = e− x2
,
⎫
⎬⎪
⎭⎪
and periodic boundary conditions.
u(x,t) =
e3t
2 0.8πte−(ξ2 )e
(−(x−1.5t−ξ)2
4⋅0.8t)
−10
10
∫ dξ.
The exact solution of this problem is the following equation:
Test problem in order to know the order of the method ADER-FV:
ADER-FV
Convective flux WENO-GFORCE 5 th order.
Diffusive flux Centered reconstruction 6 th order.
Reactive term Centered reconstruction 6 th order.
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem. 3.3. Scalar problem of
LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
14
( ) ( ) ( ) ( )2
2, 0.1 , , 0.1 , , [-10,10],
5 si x 0,( ,0)
0.5 si x > 0.L
R
u x t u x t K u x t u x t xt x x
uu x
u
⎫∂ ∂ ∂+ − = ∈ ⎪∂ ∂ ∂ ⎪
⎬= ≤⎧ ⎪= ⎨ ⎪=⎩ ⎭
u(x,t) =
e0.1t
2 Kπt5e
(−(x−0.1t−ξ)2
4⋅Kt)
−10
0
∫ dξ+ 0.5e(−
(x−0.1t−ξ)2
4⋅Kt)
0
10
∫ dξ.
The exact solution is
We can assume that u(x,t) is a solute concentration that flows inside a porous and saturated geological unit, due to advective-reactive-diffusive mechanisms. In order to describe this, we have the following transport equation, in which we want to observe the effect of variation in the Diffusion coefficient:
u(−10,t) = 5e0.1t ,u(10,t) = 0.5e0.1t .
⎧⎨⎪
⎩⎪Dirichlet boundary
conditions
Parameters:
Courant = 0.4; Dif. Coef. (α) = 0.4; Time = 10;
Δx = 1/15; Δt = min(coΔx/0.1, αΔx2/K).
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem. 3.3. Scalar problem of
LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
15R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem. 3.3. Scalar problem of
LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.
Spatial Coordinate (x)
Solu
tion:
u(x
,10)
Exact sol. K=0Num. sol. K=0Exact sol. K=0.1Num. sol K=0.1Exact sol. K=0.25Num. sol. K=0.25 Exact sol. K=0.5Num. sol. K=0.5
16
4. Conclusions and future research.
The results obtained for the ADER-FV and ADER-DG suggest that is possible to reach high order for smooth solutions and avoid spurious oscillations near discontinuities. We also observe improvements in the solution when we move from the lower to higher order schemes.
It can be noted that we reach the theoretical first order accuracy in the integral L1 norm, for discontinuous solutions.
It is difficult to obtain accurate solutions near discontinuities.
When dealing with stiff source terms, classical ADER-FV may not work, while the new ADER-DG does a good job for these cases. Also we avoid the use of Taylor expansion in time and we do not need to use the C-K procedure (especially cumbersome in the case of nonlinear problems).
The new ADER-DG can be considered as more general, since it can be implemented for a general physical flux so that, if we change the problem we are solving, we just have to plug in the scheme the new flux.
Further research will include the extension to 2D and 3D, as well as resolution of Geological Engineering problems.
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009
R. Castedo, A. HidalgoDpto. de Matemática Aplicada y Métodos Informáticos,
E.T.S.I. Minas, Univ. Politécnica de [email protected], [email protected]
R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009