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Journal ArticleDOI

A high-order finite volume method for systems of conservation laws-Multi-dimensional Optimal Order Detection (MOOD)

01 May 2011-Journal of Computational Physics (Academic Press)-Vol. 230, Iss: 10, pp 4028-4050
TL;DR: Numerical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of the multi-dimensional Optimal Order Detection approach.
About: This article is published in Journal of Computational Physics.The article was published on 2011-05-01 and is currently open access. It has received 267 citations till now. The article focuses on the topics: Finite volume method & Polynomial.

Summary (4 min read)

1 Introduction

  • High-order methods for systems of nonlinear conservation laws are an important challenging question with a wide range of applications.
  • Furthermore in an engineering context such methods may deal with complex multi-dimensional domains requiring unstructured, heterogeneous or even non-conformal meshes.
  • MUSCL methods are probably the most popular second-order finite volume schemes.
  • Section 2 is dedicated to the generic framework used to describe the MOOD method.
  • The MOOD method for scalar problems is detailed in the fourth section while section 5 is dedicated to an extension of MOOD method to the Euler equations.

2 General framework

  • Boundary conditions shall be prescribed in the following.
  • To elaborate the discretization in space and time, the authors introduce the following ingredients.
  • The authors assume that the computation domain Ω is a polygonal bounded set of R2 divided into quadrangles Ki, i ∈ Eel where Eel is the cell index set with ci being the cell centroid.
  • Note that a high-order scheme in space and time can be rewritten as convex combinations of the first-order scheme.
  • To this end, the authors state the following definition.

3 A short review on a multi-dimensional MUSCL method

  • All L∞ stable second-order schemes are based on piecewise linear reconstructions equipped with a limiting procedure.
  • The polynomial reconstruction provides the accuracy while the limitation algorithm ensures the physical relevancy of the numerical approximation.
  • The authors briefly present the piecewise linear reconstruction step and recall the MLP method proposed in [21] which is used in the numerical part of this paper.

3.1 Linear reconstruction

  • Usually the authors ask for the following criteria Criterion 3.
  • The first condition of criterion 3 is directly satisfied and classical techniques like least squares methods are used to determine vector G that minimizes the functional E in equation (8) .

3.2 Gradient limitation

  • As the authors mentioned above, a finite volume scheme only based on a local polynomial reconstruction without limiting procedure produces spurious oscillations.
  • The authors choose to detail and use the MLP limiter instead of the classical Barth-Jespersen limiter because it provides more accurate results (see [21]).
  • The MLP technique provides a second-order finite volume scheme which satisfies the Discrete Maximum Principle under a more restrictive CFL condition than the CFL condition of the first-order scheme.
  • The extension of MUSCL type methods to piecewise quadratic or even higher degree polynomials in a multi-dimensional context is not yet achieved.

4 The Multi-dimensional Optimal Order Detection method (MOOD)

  • Classical high-order methods are based on an a priori limitation of the reconstructed values which are plugged into a one time step generic finite volume scheme to update the mean values .
  • Unlike existing methods, the MOOD technique proceeds with an a posteriori limitation.
  • Then the a posteriori limitation consists of reducing the polynomial degree and recomputing the predicted solution u⋆h until the DMP property (7) is achieved.
  • To this end, a prescribed maximum degree dmax is introduced and used to perform an initial polynomial reconstruction on each cell.
  • In the following the authors focus on the quadratic polynomial case dmax = 2 and first present the local quadratic reconstruction of [19].

4.2 Description of the MOOD method

  • The authors now detail the MOOD technique considering the simple case where an explicit time discretization is employed.
  • Moreover, without loss of generality, the authors present the method using only one quadrature point (R = 1) and skip the subscript r denoting uij in place of uij,r.
  • To this end, the authors define the following fundamental notions.
  • Di is the Cell Polynomial Degree which represents the degree of the polynomial reconstruction on cell Ki. dij and dji are the Edge Polynomial Degrees which correspond to the effective degrees used to respectively build uij and uji on both sides of edge eij .

4.3 Convergence of the MOOD method

  • The authors first recall the classical stability result (see [6] and references herein).
  • Let us consider the generic first-order finite volume scheme (2) with reflective boundary conditions.
  • If the numerical flux is consistent and monotone, then the DMP property given by definition 2 is satisfied.
  • If the EPD strategy is upper-limiting then the MOOD method provides an updated solution un+1h which satisfies the DMP property after a finite number of iterations.
  • Then there is at least one cell having its CellPD positive which has to be decremented.

5 Extension to the Euler Equations

  • Classical methods such as the MUSCL technique use a limiting procedure derived from the scalar case to keep the numerical solution from producing spurious oscillations.
  • The authors thus propose a strategy to both have an accurate approximation where the solution is smooth and prevent the oscillations from appearing close to the discontinuities.
  • In that case, first-order values are substituted to the unphysical reconstructed values, for instance if the reconstructed value ρij is negative on cell Ki, the authors replace it with the mean value ρi.
  • Instead of using one CellPD per cell and per variable, the authors choose to define only one CellPD per cell and to use it for all variables.
  • Next section is dedicated to numerical experiments to assess the computational efficiency of the MOOD method.

6 Numerical results — the scalar case

  • Notice that the authors use ν(i) as the reconstruction stencil.
  • Following remark 12, the authors simply apply the MOOD procedure detailed in section 4.2 to each sub-step of the RK3-TVD.
  • The CellPD are thus reinitialized to dmax at the beginning of each time sub-step.

6.1 Test descriptions

  • Two classical numerical experiments are carried out to demonstrate the ability of the method to provide effective third-order accuracy and to handle discontinuities with a very low numerical diffusion.
  • Since the authors use periodic boundary conditions, the final time corresponds to a full revolution such that the exact solution coincides with the initial one.
  • First introduced by R.J. Leveque in [17], this solid body rotation test uses three shapes which are a hump, a cone and a slotted cylinder.

6.2 Numerical results

  • 2.1 Comparison between EPD1 and EPD2 strategies Figure 3 shows that the optimal convergence rate in L1 error for P1, MOODP1 and MLP methods is achieved since the curves fit very well.
  • The smaller number of isolines outside of the slot is, the more accurate the scheme is.
  • 2.3 Comparison between FV, MLP, MOOD-P1 and MOOD-P2 with EPD1 strategy on non-uniform meshes Approximation accuracy is reduced when one employs meshes with large deformations, i.e. the elements are no longer rectangular but quadrilateral with large aspect ratios.
  • The authors notice that MOOD-P1 gives a clearly better solution than the one computed with MLP, even on the smooth profiles.

7 Numerical results — the Euler case

  • The authors now turn to the Euler equations (16) to test the MOODmethod.
  • In the present article, the authors use the HLL numerical flux detailed in [26].
  • Sod shock tube is used to test the ability of MOOD in reproducing simple waves.
  • These two tests are run with MLP, MOOD-P1 and MOOD-P2 on uniform meshes for comparison purposes with classical results from literature.

7.1 Sod Shock Tube

  • The one dimensional Sod problem is used as a sanity check for the MOOD method.
  • The exact solution is invariant in x2-direction.
  • However the authors observe an undershoot (resp. overshoot) at the tail of the rarefaction with MOOD-P2 for the density (resp. velocity).
  • The density and the x1-velocity solutions at the final time using the MLP, MOOD-P1 and MOOD-P2 methods are also printed in figure 9.
  • The MOOD-P1 is an intermediate case where the dispersion is reduced in comparison with the MLP method but where the MOOD-P2 accuracy is not reached.

7.3 Mach 3 wind tunnel with a step

  • The wind tunnel is 1 length unit wide and 3 length units long and the step is located at 0.6 length unit from the left-hand side of the domain.
  • An inflow condition is set on the left boundary and an outflow condition on the right one.
  • The authors first consider the situation with coarse mesh using 120× 40 cells.
  • With the MLP method, the authors remark that the formation of a triple point at x1 = 1.25 above the step (at a distance of about 0.1) while the junction point should be exactly on the step interface.
  • Nevertheless MOOD methods still provide the best numerical approximations.

7.4 Double Mach reflection of a strong shock

  • The last problem is the double mach reflection of a strong shock proposed in [29].
  • Then zoomed top views of 50 isolines — between minimal and maximal values, ρm and ρM respectively, taken over the results of the three methods on a same mesh — of the results obtained with the HLL flux are plotted in figure 14 for the 960× 240 uniform mesh on left and for the 1920× 480 one on right.
  • The first Mach stem M1 is connected to the main triple junction point with the incident shock wave and the reflected wave.
  • Indeed, the application of a strict DMP reduces the accuracy of the scheme in the vicinity of the slip line maintaining a too large amount of diffusion.
  • The MOOD-P2 computational cost is competitive (at most around 2.7 times more expensive than MLP on their numerical experiments) in regard to the observed accuracy improvement, see for instance figures 7 or 9.

8 Conclusion and perspectives

  • This paper presents a high-order polynomial finite volume method named Multi-dimensional Optimal Order Detection (MOOD) for conservation laws.
  • Contrarily to classical high-order methods MOOD procedure is based on a test of the Discrete Maximum Principle (DMP) after an evaluation of the solution with unlimited polynomials.
  • The MOOD method is an a posteriori limiting process, whereas classical limiting strategies perform an a priori limitation.
  • The MOOD method has no restriction to deal with higher polynomial degrees and polygonal meshes.
  • Two-dimensional numerical results are provided for advection and the Euler equations problems on regular and highly non-regular quadrangular meshes.

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Frequently Asked Questions (1)
Q1. What are the contributions in "A high-order finite volume method for systems of conservation laws — multi-dimensional optimal order detection (mood)" ?

In this paper, the authors investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Numerical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of this approach.