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European Journal of Applied Sciences – Vol. 10, No. 1

Publication Date: February 25, 2022

DOI:10.14738/aivp.101.11414. Remaki, L. (2022). Waves Speed Averaging Impact on Godunov Type Schemes for Hyperbolic Equations with Discontinuous

Coefficients: The Linear Scalar Case. European Journal of Applied Sciences, 10(1). 333-347.

Services for Science and Education – United Kingdom

Waves Speed Averaging Impact on Godunov Type Schemes for

Hyperbolic Equations with Discontinuous Coefficients: The

Linear Scalar Case

L. Remaki

BCAM - Basque Centre for Applied Mathematics

Mazarredo, 14. 48009 Bilbao Basque Country – Spain

Department of mathematics ad computer science

Alfaisal University, KSA

ABSTRACT

This paper deals with the wave speed averaging impact on Godunov type schemes

for linear scalar hyperbolic equations with discontinuous coefficients. In many

numerical schemes of Godunov type used in fluid dynamics, electromagnetic,

electro-hydrodynamic problems and so on, a Riemann problem needs to be solved

to estimate fluxes. The exact solution is generally not possible to obtain, but good

approximations are available, the more popular being Roe and HLLC Riemann

solvers. However, all these solvers assume that the acoustic waves speeds are

continuous by considering some averaging. This could unfortunately lead to a

wrong solution as we will show in this paper for the linear scalar case. Providing a

Riemann solver in the general case of non-linear hyperbolic systems with

discontinuous waves speed is a very hard task, therefore in this paper and as a first

step, we focus on the linear and scalar case. In a previous work a Riemann solver

that considers waves speed discontinuities was proposed. A numerical argument to

show the validity of the solution was provided. In this paper, first a new argument

is developed using a regularization technique, to reinforce the validity of the

proposed solution. Then, the corresponding Godunov scheme is derived, and the

effect of waves speed averaging is demonstrated with a clear connection to the

distribution product phenomenon.

Keyword: Hyperbolic equations, Riemann solver, waves speed, Godunov scheme, CFD,

generalized functions

INTRODUCTION

In many numerical methods such as finite volume, Discontinuous Galerkin (DG), Discontinuous

finite volume, and so on, estimation of numerical fluxes at cells (sub-cells) faces is a crucial part

of the numerical scheme. The accuracy of the method depends on the accuracy of the flux

estimation. For the convective fluxes, generally a Riemann problem is considered and then an

approximation Riemann solver is used. This leads to a stable upwind numerical schemes. This

approach was first proposed by Godunov [1], consequently such methods are referred to by

Godunov type methods. Depending of the problem to solve, many Riemann solver

approximations where developed. Among the most popular in computational fluid dynamics,

we can cite the Roe solver [2,3], the HLL Riemann solver [7] and the HLLC solver [6]. For the

Roe solver, the Jacobian matrix is averaged in such a way that hyperbolicity, consistency with

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022

Services for Science and Education – United Kingdom

the exact Jacobian and conservation across discontinuities still fulfilled. This solver has been

modified [4,5], to overcome the shortcoming for low-density flows. The HLL solver, solves the

original nonlinear flux to take nonlinearity into account. It has a major drawback however

because of space averaging process, the contact discontinuities, shear waves and material

interfaces are not captured. To remedy this problem, the HLLC solver was proposed by adding

the missing wave to the structure. However all these methods assume that the waves speed are

continuous across the left and right states of the Riemann problem (through the cell interfaces

of the mesh) by applying diverse averaging process. This is not true in general; typical situations

are recirculation for turbulent flows and transitions from subsonic to supersonic for transonic

regimes. The impact of this averaging on the obtained numerical methods has never been

addressed. To handle this important issue, a Riemann solver of scalar hyperbolic linear

equation with discontinuous coefficient is developed and published in a proceeding [8]

providing a numerical argument of its validity, this was based on a first idea developed in [11].

This solver takes into account the discontinuities of waves speed. To reinforce the validity of

the proposed solver and in absence of a rigorous mathematical proof, another strong argument

based on a regularization strategy is proposed in this work. Then, and to assess the impact of

the averaging process, a Godunov type scheme is derived using the proposed Riemann solver

and results are compared to the case of averaged waves speed. The tests show as well that the

critical situation corresponds to the case where a product of distributions occurs, which is not

defined by the classical theory of distributions, which could explain the non-validity of the

waves speed averaging.

RIEMANN SOLUTION FOR HYPERBOLIC EQUATION WITH DISCONTINUOUS COEFFICIENT

Consider the following scalar linear hyperbolic equation with discontinuous coefficient,

!

!"

� + �(�) !

!# � = 0, on [�, �] × �

�(0, �) = �$ ∈ �%(�)

�(. ) ∈ �%(�)

(1)

The initial condition �$(�) and the coefficient �(�) are only bounded functions and can be

discontinuous. From the theoretical point of view, the well-posedness of this type of problems

is studied in [9]. It is shown that the more critical case is when the solution � and the coefficient

�(. ) are discontinuous at the same location which leads to a product of distributions (for

instance if �(. ) is some Heaviside function and a Dirac function resulting from the derivative of

�). This product is not defined in the classical space of distributions which is not an algebra.

The well-posdeness of the problem is then studied in a more appropriate space of generalized

functions introduced by J.F Colombeau, known as well as the Colombeau’s algebra. For more

details we refer to [9,13,14]. In the numerical tests section, the impact of the product of

distributions issue on numerical results is demonstrated.

Now, let’s define the Riemann problem associated with problem (1)

!

!"

� + �(�) !

!# � = 0, on [�, �] × �

�(0, �) = �$ = 4

�& if � < 0

�' if � > 0

�(�) = 4

�& if � < 0

�' if � > 0.

(2)

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Remaki, L. (2022). Waves Speed Averaging Impact on Godunov Type Schemes for Hyperbolic Equations with Discontinuous Coefficients: The Linear

Scalar Case. European Journal of Applied Sciences, 10(1). 333-347.

URL: http://dx.doi.org/10.14738/aivp.101.11414

In this equation, the acoustic wave speed �(. ) is discontinuous, which again, is not taken into

account in the existing Riemann solvers where acoustic waves speed is averaged. In [8], a

Riemann solver is proposed based on the following observations of different possible

situations:

Case 1: �& > 0 and �' > 0 we have propagation of the discontinuity (of initial condition) to the

right and we do not need to consider what happening within the fan defined by the two acoustic

waves, because they will catch up if �& > �' and if �& < �' an expansion will appear.

Case 2: �& < 0 and �' < 0 similar to the previous case with a propagation of the discontinuity

to the left.

Case 3: �& < 0 and �' > 0 we have propagation of the discontinuity to the left and the right

simultaneously, and we need to determine what happened within the fan defined by the two

acoustics waves. We assume that a constant state appears and its expression will be given

below.

Case 4: �& > 0 and �' < 0 in this case we have opposite acoustic waves speed and then the

discontinuity will remain blocked, which means there is no propagation.

Based on the above observations, the Riemann solution of problem (2) is given by

�(�,�) = 9

�& if �& > 0 ��� �' > 0

� if �& < 0 ��� �' > 0

�' if �& < 0 ��� �' < 0

�$ if �& > 0 ��� �' < 0

(3)

Where the expression of the constant � is given by

� =

!

|#$|

($) !

|#%|

(%

!

|#$|

) !

|#%|

(4)

ON THE PROOF

In this section we will first recall the numerical argument presented in [8] with more details

and then we propose another strong argument that supports the validity of the proposed

Riemann solver. This argument is based first on a regularization strategy that allows computing

the exact solution using characteristics technique and finally obtain the exact solution of the

original problem as an asymptotic limit of the regularized solution.

Numerical argument

To prove the validity of solution (3) and formula (5) at least numerically, the Riemann problem

(2) is solved using a centered second order finite volume scheme stabilized by a first order

artificial viscosity, which is equivalent in this case to a finite difference scheme. Let’s set

�*

+ = �(�+, �*) �*

+ = �(�+, �*)

Then the explicit numerical scheme is give by

�*

+), = �*

+), + �*

+�� >

�*), + − �*-, +

2h B + ��� >

�*), + − 2�*

+ + �*-, +

h. B