Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
On order conditions of IMEX R-K schemes applied
to hyperbolic systems with diffusive relaxation
S. Boscarino1 , L. Pareschi2 , G. Russo1
1 Department
of Mathematics and Computer Science
University of Catania,
[email protected], [email protected]
2 Department of Mathematics
University of Ferrara
[email protected]
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Outline
1
Introduction
What is an IMEX R-K scheme?
2
IMEX R-K schemes for hyperbolic systems with diffusive
relaxation
Diffusive relaxation
3
Two different approaches
IMEX-I Approach
IMEX-E Approach
4
Numerical Results
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Introduction
Time Discretization, IMEX R-K Schemes
We consider initial value problems for systems of ordinary
differential equations (ODEs) of the form
y 0 = f (y ) + g (y ),
y (x0 ) = y0 ,
where f , g : Rk → Rk , are sufficiently smooth functions with
different stiffness properties. Such system may arise from the
spatial discretization of a system of PDEs.
For the numerical integration of such system we use additive
Runge-Kutta schemes.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Introduction
What is an IMEX R-K scheme?
The idea is to consider two different s-stage Runge-Kutta scheme,
c̃
Ã
c
A
.
b̃
T
bT
and use one of them for the function f (Ã, b̃), and the other one for
the function g , (A, b)
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Introduction
What is an IMEX R-K scheme?
The idea is to consider two different s-stage Runge-Kutta scheme,
c̃
Ã
c
A
.
b̃
T
bT
and use one of them for the function f (Ã, b̃), and the other one for
the function g , (A, b)
Additive Runge Kutta schemes combining implicit and explicit
schemes are known in the literature as IMplicit-EXplicit (IMEX)
Runge-Kutta (RK) schemes.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
What is an IMEX R-K scheme?
IMEX R-K schemes
An (IMEX) R-K scheme has the form
Yi
y1
=
y0 + h
= y0 + h
i−1
X
j=1
ν
X
ãij f (t0 + c̃j h, Yj ) + h
b̃i f (t0 + c̃i h, Yi ) + h
i=1
ν
X
j=1
ν
X
aij g (t0 + cj h, Yj ),
bi g (t0 + ci h, Yi ).
i=1
à = (ãij ), ãij = 0, j ≥ i and A = (aij ): ν × ν matrices.
Coefficient vectors: c̃ = (c̃1 , . . . , c̃ν )T , b̃ T = (b̃1T , . . . , b̃νT )T ,
c = (c1 , . . . , cν )T , b = (b1 , . . . , bν )T .
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
What is an IMEX R-K scheme?
IMEX R-K schemes
The IMEX RK scheme (Ã, b̃) and (A, b) is chosen with the
aim of efficiently integrating the previous system with low
computational cost. For example, if f represents the nonstiff
part of the system and g the stiff part of it, an explicit
method can be used for f and an implicit one for g.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
What is an IMEX R-K scheme?
IMEX R-K schemes
The IMEX RK scheme (Ã, b̃) and (A, b) is chosen with the
aim of efficiently integrating the previous system with low
computational cost. For example, if f represents the nonstiff
part of the system and g the stiff part of it, an explicit
method can be used for f and an implicit one for g.
Sufficient condition to guarantee that f is always evaluated
explicitly: the scheme for g is diagonally implicit (DIRK ).
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Classical Order conditions
We report the order conditions for IMEX R-K schemes up to order p = 3. We assume
that the coefficients ci , c̃i , aij , ãij satisfy conditions
P
P
c̃i = j ãi,j , ci = j ai,j ,
then the order conditions are:
First order:
P
Second order:
P
i b̃i c̃i = 1/2,
Third order:
P
ij b̃i ãij c̃j = 1/6,
P
i
ij
P
bi ci = 1/2,
P
b̃i c̃i c̃i = 1/3,
P
i
P
i
bi ãij cj = 1/6,
b̃i ci ci = 1/3,
P
P
Coupling conditions:
P
ij b̃i ãij cj = 1/6,
P
b̃i = 1,
i
i
P
P
i
bi = 1.
i
b̃i ci = 1/2,
ij
bi aij cj = 1/6,
ij
b̃i aij c̃j = 1/6,
P
ij
bi aij c̃j = 1/6,
P
b̃i c̃i ci = 1/3,
S. Boscarino
P
i
P
i
bi c̃i = 1/2,
P
i
bi ci ci = 1/3,
ij
b̃i aij cj = 1/6,
ij
bi ãij c̃j = 1/6,
bi c̃i c̃i = 1/3,
P
i bi c̃i ci = 1/3.
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
The number of coupling conditions increase dramatically with the
order of the schemes.
IMEX-RK
order
1
2
3
4
5
6
Number of coupling conditions
General case b̃i = bi c̃ = c c̃ = c and b̃i = bi
0
0
0
0
2
0
0
0
12
3
2
0
56
21
12
2
252
110
54
15
1128
528
218
78
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Classification of IMEX R-K schemes
Definition 1 ( Methods of Type A) The matrix A is
invertible.
Definition 2 ( Methods of Type CK)
0 0
A=
a Â
The submatrix  is invertible.
Definition 3 ( Methods of Type ARS) Special case of type
CK with vector a = 0 and submatrix  invertible.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
What is an IMEX R-K scheme?
Some References:
U. Asher, S. Ruuth, and R. J. Spiteri: Implicit-explicit Runge-Kutta methods for
time depended Partial Differential Equations. Appl. Numer. Math. 25, (1997),
pp. 151-167.
C. A. Kennedy, M. H. Carpenter: Additive Runge-Kutta schems for
convection-diffusion-reaction equations. Appl. Numer. Math. 44 (2003), no.
1-2, 139–181.
L.Pareschi, G. Russo: Implicit-Explicit Runge-Kutta schemes and Applications
to hyperbolic systems with relaxation. Journal of Scientific Computing Volume:
25, Issue: 1, October, 2005, pp. 129-155: theory, numerics, applications,
241–251, Springer, Berlin, 2003.
X. Zhong: Additive semi-implicit Runge-Kutta methods for computing
high-speed nonequilibrium reactive flows. J. Comput. Phys., 20 (1996), pp.
191-209.
S. B.:On an accurate third order implicit-explicit RungeKutta method for stiff
problems, Applied Numerical Mathematics, 59 (2009), pp. 15151528.
S.B., G. Russo:On a class of uniformly accurate IMEX Runge-Kutta schemes
and application to hyperbolic systems with relaxation. SIAM J. SCI. COMPUT.
Vol. 31, (2009), No. 3, pp. 1926-1945
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
1
Introduction
What is an IMEX R-K scheme?
2
IMEX R-K schemes for hyperbolic systems with diffusive
relaxation
Diffusive relaxation
3
Two different approaches
IMEX-I Approach
IMEX-E Approach
4
Numerical Results
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Diffusive relaxation
A simple prototype of hyperbolic system with relaxation term is given by:
∂τ u + ∂ξ V
=
0,
1
= − (V − Q(u))
ε
M
where u = u(x, τ ), V = V (x, τ ) ∈ R , ε > 0 is called the relaxation
time. Under the rescaling (diffusive scaling)
∂τ V + ∂ζ p(u)
τ = t/ε,
ξ = x,
V = εv ,
q(u) = Q(u)/ε,
we obtain a general diffusive relaxation system given by:
∂t u + ∂x v = 0,
1
1
∂t v + 2 ∂x p(u) = − 2 (v − q(u))
ε
ε
Where p 0 (u) > 0. This system is hyperbolic with two distinct real
p
characteristics speed p 0 (u)/ε.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Diffusive relaxation
In the small relaxaction limit, ε → 0 the system relax towards the system
∂t u + ∂x q(u) = ∂xx p(u),
v = q(u) − ∂x p(u).
Since the equilibrium equation is of parabolic type, the main stability
condition for the diffusive relaxation system is
|q 0 (u)|2 <
p 0 (u)
ε2
and it is naturally satisfied in the limit ε → 0.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
S. Boscarino
1
ε
leading to a CFL
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
1
ε
leading to a CFL
Construct numerical schemes with the correct asymptotic limit.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
1
ε
leading to a CFL
Construct numerical schemes with the correct asymptotic limit.
The resulting scheme, in the limit ε → 0 converges to an explicit scheme for the
diffusion equation, with the usual CFL parabolic condition ∆t ≈ ∆x 2 .
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
1
ε
leading to a CFL
Construct numerical schemes with the correct asymptotic limit.
The resulting scheme, in the limit ε → 0 converges to an explicit scheme for the
diffusion equation, with the usual CFL parabolic condition ∆t ≈ ∆x 2 .
Some References:
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
1
ε
leading to a CFL
Construct numerical schemes with the correct asymptotic limit.
The resulting scheme, in the limit ε → 0 converges to an explicit scheme for the
diffusion equation, with the usual CFL parabolic condition ∆t ≈ ∆x 2 .
Some References:
G. Naldi, L. Pareschi: Numerical Schemes for hyperbolic systems of
conservation laws with stiff Diffusive relaxation. SIAM. J. Num. Anal. Vol. 37,
No. 4 (2000), pp. 1246-1270.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Motivations
Most of the popular methods fail to capture the correct behaviour of the
solution in the relaxation limit (ε → 0)
stiffness both in the convection and in the relaxation terms.
characteristic speed of the hyperbolic part is of order
condition like ∆t ≈ ε∆x.
1
ε
leading to a CFL
Construct numerical schemes with the correct asymptotic limit.
The resulting scheme, in the limit ε → 0 converges to an explicit scheme for the
diffusion equation, with the usual CFL parabolic condition ∆t ≈ ∆x 2 .
Some References:
G. Naldi, L. Pareschi: Numerical Schemes for hyperbolic systems of
conservation laws with stiff Diffusive relaxation. SIAM. J. Num. Anal. Vol. 37,
No. 4 (2000), pp. 1246-1270.
S. Jin, L. Pareschi and G. Toscani: Diffusive relaxation for multiscale
Discrete-Velocity Kinetic Equations. SIAM. J. Num. Anal. Vol. 35, No. 6
(1998), pp. 2405-2439.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Goal of the present work
To overcome the CFL parabolic restriction obtaining IMEX R-K
schemes that have as CFL condition ∆t ≈ ∆x , with corse grid ∆t,
∆x >> ε.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Goal of the present work
To overcome the CFL parabolic restriction obtaining IMEX R-K
schemes that have as CFL condition ∆t ≈ ∆x , with corse grid ∆t,
∆x >> ε.
To introduce two different approachs.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Goal of the present work
To overcome the CFL parabolic restriction obtaining IMEX R-K
schemes that have as CFL condition ∆t ≈ ∆x , with corse grid ∆t,
∆x >> ε.
To introduce two different approachs.
To perform the analysis of the behaviour of IMEX RK schemes
when ε → 0, for the two different approachs.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Goal of the present work
To overcome the CFL parabolic restriction obtaining IMEX R-K
schemes that have as CFL condition ∆t ≈ ∆x , with corse grid ∆t,
∆x >> ε.
To introduce two different approachs.
To perform the analysis of the behaviour of IMEX RK schemes
when ε → 0, for the two different approachs.
To have guidelines for the construction of high order schemes.
To this purpose we reformulate the previous diffusive relaxation
system such that it allows us to design a class of IMEX
Runge-Kutta (RK) schemes that in the diffusive limit no diffusive
restriction is appears on the time step.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Diffusive relaxation
Diffusive relaxation
Overcoming parabolic stiffness
The starting point is to consider the system
∂t u = −∂x v ,
2
ε ∂t v
= −∂x p(u) − (v − q(u)).
We add and subtract the term µ(ε)∂xx p(u) and consider the
equivalent system
∂t u = −∂x (v + µ∂x p(u)) + µ∂xx p(u),
2
ε ∂t v
= −∂x p(u) − (v − q(u)).
µ(ε) is such that µ : R+ → [0, 1], µ(0) = 1 and µ(1) = 0.
We present two different approach in order to solve numerically the
previous system in the diffusive limit, i.e. ε → 0
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
Two different approaches
IMEX-I Approach
First approach called IMEX-I approach‡
∂t u = −∂x (v + µ∂x p(u)) + µ∂xx p(u),
|
{z
} | {z }
explicit
2
ε ∂t v
implicit
= −∂x p(u) − (v − q(u))
{z
}
|
implicit
S. B. , L. Pareschi, G. Russo‡ : Implicit-Explicit Runge-Kutta
schemes for hyperbolic systems and kinetic equations in the
diffusion limit. submitted to SIAM J. SCI. COMPUT.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
In the IMEX-I approach the diffusive system can be written in the form
u0
=
2 0
=
ε v
f1 (u, v ) + f2 (u),
g (u, v ).
where f1 (u, v ) = −∂x (v + µ∂x p(u)), f2 (u) = µ∂xx p(u) and
g (u, v ) = (−∂x p(u) − v + q(u)).
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
In the IMEX-I approach the diffusive system can be written in the form
u0
=
2 0
=
ε v
f1 (u, v ) + f2 (u),
g (u, v ).
where f1 (u, v ) = −∂x (v + µ∂x p(u)), f2 (u) = µ∂xx p(u) and
g (u, v ) = (−∂x p(u) − v + q(u)).
When ε → 0 we get
u 0 = fˆ1 (u) + f2 (u),
0
= g (u, v ). (MANIFOLD M = {(u, v ) ∈ R|g (u, v ) = 0})
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
In the IMEX-I approach the diffusive system can be written in the form
u0
=
2 0
=
ε v
f1 (u, v ) + f2 (u),
g (u, v ).
where f1 (u, v ) = −∂x (v + µ∂x p(u)), f2 (u) = µ∂xx p(u) and
g (u, v ) = (−∂x p(u) − v + q(u)).
When ε → 0 we get
u 0 = fˆ1 (u) + f2 (u),
0
= g (u, v ). (MANIFOLD M = {(u, v ) ∈ R|g (u, v ) = 0})
i.e. when ε → 0 the numerical solution is projected onto the
manifold g (u, v ) = 0 (v = G (u), assumed that the Jacobian matrix
gv (u, v ) is invertible). with fˆ(u) = f (u, G (u)). Previous system is
called a REDUCED SYSTEM.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Definition
We say that an IMEX R-K scheme is globally stiffly accurate, GSA, if
1
The implicit R-K scheme is stiffly accurate, SA, if esT A = b T , with
esT = (0, ..., 0,
1
). This property is important for the
|{z}
sth−comp.
L-stability of the scheme.
2
The s-stage explicit R-K scheme satisfies the condition esT Ã = b̃ T
(FSAL, First Same As Last ).
cs = c̃s = 1, i.e. the numerical solution is identical to the last
internal stage values of the scheme.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Definition
We say that an IMEX R-K scheme is globally stiffly accurate, GSA, if
1
The implicit R-K scheme is stiffly accurate, SA, if esT A = b T , with
esT = (0, ..., 0,
1
). This property is important for the
|{z}
sth−comp.
L-stability of the scheme.
2
The s-stage explicit R-K scheme satisfies the condition esT Ã = b̃ T
(FSAL, First Same As Last ).
cs = c̃s = 1, i.e. the numerical solution is identical to the last
internal stage values of the scheme.
Initial conditions (u0 , v0 ) are well-prepared if g (u0 , v0 ) = 0 .
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Analysis of TYPE A IMEX R-K scheme (for ε → 0).
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Analysis of TYPE A IMEX R-K scheme (for ε → 0).
if such scheme is globally stiffly accurate, g (Us , Vs ) = 0, i.e. the
last stages lie on the manifold then un+1 = Us and vn+1 = Vs and
g (un+1 , vn+1 ) = 0
S. Boscarino
i.e. vn+1 = G (un+1 ) .
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Analysis of TYPE A IMEX R-K scheme (for ε → 0).
if such scheme is globally stiffly accurate, g (Us , Vs ) = 0, i.e. the
last stages lie on the manifold then un+1 = Us and vn+1 = Vs and
g (un+1 , vn+1 ) = 0
i.e. vn+1 = G (un+1 ) .
Analysis of TYPE CK IMEX R-K scheme (ε → 0).
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
Analysis of TYPE A IMEX R-K scheme (for ε → 0).
if such scheme is globally stiffly accurate, g (Us , Vs ) = 0, i.e. the
last stages lie on the manifold then un+1 = Us and vn+1 = Vs and
g (un+1 , vn+1 ) = 0
i.e. vn+1 = G (un+1 ) .
Analysis of TYPE CK IMEX R-K scheme (ε → 0).
If the following condition is satisfied
T
αs = −ês−1
Â−1 a = 0
then g (Us , Vs ) = 0, i.e. the last stages lie on the manifold, and if
the scheme is globally stiffly accurate we obtain un+1 = Us and
v n + 1 = Vs and
g (un+1 , vn+1 ) = 0 , i.e. vn+1 = G (un+1 ) .
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-I Approach
TYPE CK R-K scheme for the IMEX-I Approach.
W-P
Yes
Yes
Yes
No
No
No
αs = 0
Yes
No
Yes/No
Yes
No
Yes/No
GSA
Yes
Yes
No
Yes
Yes
No
M
Yes
Yes
No
Yes
No
No
W-P - WELL-PREPARED Initial data
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
Additional Order Conditions in the diffusive limit (i.e.
ε → 0)
In order to maintain the order accuracy in time of the scheme, by using
Taylor’s expansion for the exact and numerical solution and by comparing
them, we get the following additional order conditions for the reduced
system up to third order:
No-SA
b T A−1 c̃ = 1
b T A−1 c̃ 2 = 1
T −1
b A Ãc̃ = 1/2
SA
c̃ = 1
c̃ = 1
T
es Ãc̃ = 1/2
where c̃ = Ãe, with e = (1, ..., 1)T and es = (0, ..., 0, 1)T
If IMEX R-K scheme is GSA then the conditions are automatically
satisfied, since esT Ã = b̃ T ⇒ b̃ T c̃ = 1/2.,i.e. the classical second order
cond.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
∂t u
= −∂x (v + µ∂x p(u)) + µ∂xx p(u),
|
{z
} | {z }
∂t v
−∂x p(u) (v − q(u))
=
−
ε2 } | {z
ε2
}
| {z
explicit
explicit
implicit
implicit
We call such approach IMEX-E approach.
Second equation: p(u)x /ε2 is treated explicitly.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
∂t u
= −∂x (v + µ∂x p(u)) + µ∂xx p(u),
|
{z
} | {z }
∂t v
−∂x p(u) (v − q(u))
=
−
ε2 } | {z
ε2
}
| {z
explicit
explicit
implicit
implicit
We call such approach IMEX-E approach.
Second equation: p(u)x /ε2 is treated explicitly.
Easier to implement because fluxes have more clear interpretation.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
∂t u
= −∂x (v + µ∂x p(u)) + µ∂xx p(u),
|
{z
} | {z }
∂t v
−∂x p(u) (v − q(u))
=
−
ε2 } | {z
ε2
}
| {z
explicit
explicit
implicit
implicit
We call such approach IMEX-E approach.
Second equation: p(u)x /ε2 is treated explicitly.
Easier to implement because fluxes have more clear interpretation.
Apparently hopeless because of the diverging characteristic speeds
when ε → 0.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
∂t u
= −∂x (v + µ∂x p(u)) + µ∂xx p(u),
|
{z
} | {z }
∂t v
−∂x p(u) (v − q(u))
=
−
ε2 } | {z
ε2
}
| {z
explicit
explicit
implicit
implicit
We call such approach IMEX-E approach.
Second equation: p(u)x /ε2 is treated explicitly.
Easier to implement because fluxes have more clear interpretation.
Apparently hopeless because of the diverging characteristic speeds
when ε → 0.
S. B, G. Russo: Flux-Explicit IMEX R-K schemes for hyperbolic to
parabolic relaxation problems. submitted to SINUM
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Analysis
We set p(u) = u and q(u) = 0. We look for a Fourier solution of the
form u = û(t) exp(iξx), v = v̂ (t) exp(iξx) and inserting the ansatz into
the system and using the new variable ŵ = −i v̂ /ξ in place of v̂ the
evolution equations are
ût
ε2 ŵt
= ξ 2 (ŵ + û)−ξ 2 û,
= −û−ŵ .
Apply a TYPE A IMEX R-K scheme we obtain
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
Numerical solution
ûn+1
=
ζ 2 ŵn+1
=
ûn + θ
ζ 2 ŵn −
s
X
k=1
s
X
b̃k (Ŵk + Ûk )−θ
s
X
bk Ûk
k=1
b̃k Ûk −
k=1
s
X
bk Ŵk ,
k=1
where θ = ∆tξ 2 and ζ = ε2 /∆t. Stage values
Ûk
=
ûn + θ
k−1
X
ãkj (Ŵj + Ûj ) + θ
j=1
ζ 2 Ŵk
=
ζ 2 ŵn −
k−1
X
k
X
akj ξ 2 Ûj
j=1
ãkj Ûk −
j=1
k
X
akj Ŵj .
j=1
Solve for the stage values, insert in the numerical solution for the variable
ŵ , and obtain
ζ ŵn+1 = ζ(1 − b T A−1 1)ŵn + (b T A−1 Ã − b̃ T )Û − ζb T A−1 A−1 ÃÛ + O(ζ 2 ),
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Additional Conditions
Consistency as ζ → 0 implies
b T A−1 Ã − b̃ T = 0 .
S. Boscarino
(1)
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Additional Conditions
Consistency as ζ → 0 implies
b T A−1 Ã − b̃ T = 0 .
(1)
Furthermore, if
1 − b T A−1 e = 0
(2), =⇒ ŵn+1 = −b T A−2 ÃÛ ,
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Additional Conditions
Consistency as ζ → 0 implies
b T A−1 Ã − b̃ T = 0 .
(1)
Furthermore, if
1 − b T A−1 e = 0
(2), =⇒ ŵn+1 = −b T A−2 ÃÛ ,
Note that if the implicit scheme is SA, i.e. esT = b T A−1 , the
condition (2) is satisfied, then condition (1) is equivalent to
esT Ã = b̃ T , which means that the scheme is GSA.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Additional Conditions
Consistency as ζ → 0 implies
b T A−1 Ã − b̃ T = 0 .
(1)
Furthermore, if
1 − b T A−1 e = 0
(2), =⇒ ŵn+1 = −b T A−2 ÃÛ ,
Note that if the implicit scheme is SA, i.e. esT = b T A−1 , the
condition (2) is satisfied, then condition (1) is equivalent to
esT Ã = b̃ T , which means that the scheme is GSA.
Remark
Then a sufficient condition to guarantee that both (1) and (2) are
satisfied is that the IMEX R-K of type A is GSA.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Higher Order IMEX R-K schemes
The construction of higher order schemes that capture the
correct behavior of the solution in the limit ε → 0 is more
complicated. The condition we found in fact only prevents
divergence of the numerical solution ŵn+1 .
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Higher Order IMEX R-K schemes
The construction of higher order schemes that capture the
correct behavior of the solution in the limit ε → 0 is more
complicated. The condition we found in fact only prevents
divergence of the numerical solution ŵn+1 .
From this we derive the new additional order conditions on
the IMEX R-K schemes of type A.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Additional Conditions up to second order
By using Taylors expansion for the exact and numerical solution and by
comparing them, we get additional order conditions for the TYPE A
Additional consistency conditions: w -component
b T A−2 Ã1 = 1,
first order conditions: for u and w component
b T A−2 ÃA1 = −1,
(b T − b̃ T C)1 = 1,
where A = (ÃC − A), and C = I − A−1 Ã
Additional second order conditions: for u and w component
(b̃ T C − b T )A1 =
1
,
2
S. Boscarino
b T A−2 ÃA2 1 =
1
,
2
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Second order IMEX R-K schemes
Negative results:
Theorem
There are no second order GSA IMEX R-K schemes of type A with
three stages.
Theorem
There are no second order GSA IMEX R-K schemes of type A with
four stages where the implicit part is singly diagonally implicit
(SDIRK).
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-I Approach
IMEX-E Approach
IMEX-E Approach
Generalization.
q(u) 6= 0 and p(u) non-linear. In this case the system relaxes to a
convection-diffusion equation.
As ε → 0, the scheme becomes an IMEX R-K scheme for the limit
convection-diffusion equation, where the convection term is treated
explicitly and the diffusion term is treated implicitly.
Additional Order Conditions. We give explicitly first and second
order conditions
order 1 w − component : b T A−2 Ãc̃ = 1
order 2 u − component : b̃ T ÃA−1 c̃ = 1/2, (b T − b̃ T C)c̃ = 1/2
Of course classical order conditions, as well as new algebraic order
conditions as introduced before have to be satisfied as well.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
Convergence Results
We consider the prototype problem
∂t u + ∂x v
1
∂t v + 2 ∂x u
ε
= ∂xx u − ∂xx u,
1
= − 2 v,
ε
in the limit case this lies to the linear problem
ut (x, t) = uxx (x, t)
periodic boundary condition;
initial data: u(x, 0) = cos(x) and v (x, 0) = sin(x);
Set ε2 = 10−6 with µ(ε) = 1, final time T = 1 and ∆t ≈ ∆x. The
system is integrated for x ∈ [−π, π].
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization central difference
schemes,
S. B, G. Russo: Flux-Explicit IMEX R-K schemes for hyperbolic to
parabolic relaxation problems‡ . submitted to SINUM
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization central difference
schemes,
Second order IMEX R-K scheme AGSA(3,4,2)‡ for time
advancement,
S. B, G. Russo: Flux-Explicit IMEX R-K schemes for hyperbolic to
parabolic relaxation problems‡ . submitted to SINUM
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization central difference
schemes,
Second order IMEX R-K scheme AGSA(3,4,2)‡ for time
advancement,
The term µ(ε)p(u)xx is discretized by central differencing.
S. B, G. Russo: Flux-Explicit IMEX R-K schemes for hyperbolic to
parabolic relaxation problems‡ . submitted to SINUM
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
N
40
80
160
320
640
1280
Error
1.2014e-02
3.4484e-03
9.3359e-04
2.418e-04
6.1579e-05
1.5557e-05
Order
–
1.8007
1.8851
1.9490
1.9733
1.9849
L∞ -norm of the error and convergence rates of u in with ε2 = 10−6 , CFL
= 0.5 and ∆t = CFL∆x
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
First Numerical Test
Now we test the second order IMEX scheme to this system by solving a
Riemann problem
ρt + jx
2
ε jt + ρx
= µρxx − µρxx
= −j
Initial data
ρL
=
2.0
jL = 0, −1 < x < 0,
ρR
=
1.0
jR = 0, 0 < x < 1.
2
µ(ε) = exp(−ε /∆x),
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
First Numerical Test
Now we test the second order IMEX scheme to this system by solving a
Riemann problem
ρt + jx
2
ε jt + ρx
= µρxx − µρxx
= −j
Initial data
ρL
=
2.0
jL = 0, −1 < x < 0,
ρR
=
1.0
jR = 0, 0 < x < 1.
2
µ(ε) = exp(−ε /∆x),
we compute the scheme in the rarefied regime (ε = 0.7, µ = 0) and
in the diffusive regime (or stiff regime) for ε2 = 10−6 , µ = 1.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
First Numerical Test
Now we test the second order IMEX scheme to this system by solving a
Riemann problem
ρt + jx
2
ε jt + ρx
= µρxx − µρxx
= −j
Initial data
ρL
=
2.0
jL = 0, −1 < x < 0,
ρR
=
1.0
jR = 0, 0 < x < 1.
2
µ(ε) = exp(−ε /∆x),
we compute the scheme in the rarefied regime (ε = 0.7, µ = 0) and
in the diffusive regime (or stiff regime) for ε2 = 10−6 , µ = 1.
The boundary condition are of reflecting type. Final time t = 0.25
in the rarefied regime and t = 0.04 in the diffusive regime.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization Finite difference
schemes such us WENO schemes, e.g. third-second order
WENO(3,2) reconstruction.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization Finite difference
schemes such us WENO schemes, e.g. third-second order
WENO(3,2) reconstruction.
Second order IMEX R-K scheme AGSA(3,4,2) for time
advancement.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
We consider for the space discretization Finite difference
schemes such us WENO schemes, e.g. third-second order
WENO(3,2) reconstruction.
Second order IMEX R-K scheme AGSA(3,4,2) for time
advancement.
The term µ(ε)p(u)xx is discretized by central differencing.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
Rarefied Regime, (ε = 0.7)
2
1.9
0.5
1.8
1.7
0.4
1.6
0.3
1.5
1.4
0.2
1.3
1.2
0.1
1.1
1
−1
−0.8
−0.6
−0.4
−0.2
0
(a)
0.2
0.4
0.6
0.8
1
0
−1
−0.8
−0.6
−0.4
−0.2
0
(b)
0.2
0.4
0.6
0.8
1
Numerical solution at time t = 0.25 , in the rarefied regime (ε = 0.7)
with ∆t = 0.0025, CFL = 0.25 and ∆x = 0.01 and N = 200. From the
left to the right the mass density ρ (a) and the flow j, (b). Solid line
reference solution with N = 2000 cells.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
IMEX-E Approach−6
Parabolic Regime, (ε = 10
)
2
1.6
1.9
1.4
1.8
1.2
1.7
1
1.6
1.5
0.8
1.4
0.6
1.3
0.4
1.2
0.2
1.1
1
−1
−0.8
−0.6
−0.4
−0.2
0
(a)
0.2
0.4
0.6
0.8
1
0
−1
−0.8
−0.6
−0.4
−0.2
0
(b)
0.2
0.4
0.6
0.8
1
Numerical solution at time t = 0.04 in the parabolic regime (ε2 = 10−6 )
with ∆t = 0.001, CFL = 0.05 and ∆x = 0.02 and N = 100. From the
left to the right the mass density (a) ρ and the flow j (b). Solid line
reference solution with N = 2000 cells.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
Second Numerical Test
Finally we take the generalized Carlemann model with m = −1,
ρt
ε2 jt
∂xx ρ
∂xx ρ
− µ(ε) m
2ρm
2ρ
m
= −ρx − 2ρ j,
= −jx + µ(ε)
∂x ρ
when ε → 0, the local equilibrium is given by j = − 2ρ
m and µ(ε) → 1,
thus the system relaxes to the porous media equation.
The exact Barenblatt solution for the porous media equation,
"
2 #
1
x
2x
ρ(x, t) =
1−
, |x| < R(t),
, j(x, t) = ρ
R(t)
R(t)
R(t)3
ρ(x, t) = 0,
where R(t) = [12(t + 1)]
x ∈] − 10, 10[.
1/3
j(x, t) = 0,
|x| > R(t),
, t ≥ 0. We take ∆x = 0.2 and
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Numerical Results
Second Numerical Test
(a)
(b)
0.04
0.03
0.25
0.02
0.2
0.01
0.15
0
−0.01
0.1
−0.02
0.05
−0.03
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.04
−10
−8
−6
−4
−2
0
2
4
6
8
10
Numerical solution at time t = 3.0 for the Barenblatt problem in the
parabolic regime ε2 = 10−6 with ∆x = 0.2 , CFL = 0.5 and ∆t = 0.1.
(a) The mass density ρ and (b) the flow j.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Work in Progress
◦ Study the uniform accuracy
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Work in Progress
◦ Study the uniform accuracy
◦ Develop really high order schemes
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Work in Progress
◦ Study the uniform accuracy
◦ Develop really high order schemes
◦ Capture the Navier-Stokes limit of hyperbolic systems with
relaxation.
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic
Introduction
IMEX R-K schemes for hyperbolic systems with diffusive relaxation
Two different approaches
Numerical Results
Thank you for your attention!
S. Boscarino
On order conditions of IMEX R-K schemes applied to hyperbolic