Regarding the Numerical Diffusion of the HLLD-Type Scheme in On Energy Consistency of Intermediate States in HLL-Type MHD Riemann Solvers

Jul 16, 2026

The note addresses an issue in the HLLD-ec intermediate-state formulation for HLL-type MHD Riemann solvers, where an unphysical density oscillation appears in a slow-magnetoacoustic-shock test related to the Dai–Woodward shock tube with a strongly enhanced longitudinal magnetic field. The paper shows that a minor modification of the intermediate internal-energy expression resolves the oscillation while preserving performance on other cases. It also clarifies the dissipation/pressure-consistency implications and compares robustness against the classic HLLD scheme.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages5

Reading Time13 min

HLLD-ec energy consistency and identified oscillation issue

Modified intermediate internal energy formula

Numerical setup and validation via shock-tube tests

Dai-Woodward shock tube problem

Dai-Woodward shock tube with large Bx

Brio–Wu shock tube problem

arXiv :2607 . 10293v1 [math .NA] 11 Jul 2026 Regarding the numerical diffusion of the HLLD-type scheme in “On energy consistency of intermediate states in HLL-type MHD Riemann solvers” Fan Zhanga,b a Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, Oslo, 0315, Norway bRosseland Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern, Oslo, 0315, Norway Recently, the paper by Fan Zhang, Andrea Lani, and Stefaan Poedts,“On energy consistency of intermediate states in HLL-type MHD Riemann solvers, Journal of Computational Physics, 553 (2026), 114724”, hereafter referred to as Ref. [1], proposed a new approach to calculate the intermediate states of the HLLD scheme [2], eventually revising the diffusion term of the numerical energy flux. This newly proposed HLLD-type scheme, denoted as HLLD-ec, showed improved robustness, particularly in low plasma β test cases [1] . However, the authors were not aware of the significance of a test case in which slow magnetoacoustic shocks exhibit the major non-linearity [3] . Specifically, this test case is similar to the classic Dai-Woodward shock-tube problem [4], except that the longitudinal magnetic field Bx is enhanced by a factor of 100 . In this test case, the HLLD-ec scheme exhibits an unphysical oscillation in the density distribution, which cannot be observed in other testcases listed in Ref. [1] . It is found that this issue can be fixed by a minor change in the intermediate internal energy of the original HLLD-ec scheme, while barely affecting its performance in other test cases. Specifically, in Ref. [1], the intermediate internal energy per unit volume between the left and right going Alfvén waves was given as (ρe)** = 􀀂Sm (1 − γ) + cra*􀀃 (ρe)r* + 􀀂Sm (γ − 1) + cl*a􀀃 (ρe)l* (1) cl*a + cra* , where the superscripts l* and r* respectively denote the states behind the left and right fast magnetoacoustic waves, Sm is the propagation speed of the contact discontinuity, ca is the Alfvén speed, γ is the adiabatic index, and ρ and e are respectively density and internal energy per unit mass. Eq. (1) effectively means that the gas pressure between the left and right Alfvén waves is assumed to be constant. Instead, in this letter, we use the following formula e** = ~~ ~~􀀂Sm~~ ~~(1~~ ~~−~~ ~~γ)~~ ~~+~~ ~~cr~~a~~*􀀃~~ ~~c(ρ~~l~~*aeρ)r~~l~~**~~ ~~c􀀂S~~r~~*amρ~~r~~γ~~ ~~−~~ ~~1)~~ ~~+~~ ~~cl*~~a~~􀀃~~ ~~(ρe)l* , (2) and thus we have (ρe)l** = ρl* e** , and (ρe)r** = ρr* e** . (3) Using the new formula means that we do not force the gas pressure to change constant between the Alfvén waves, but allow it to be proportional to the intermediate densities within the Riemann fan. As the HLLD-type schemes assume that the magnetic field is constant between the Alfvén waves [2], allowing for the gas pressure to change means that the constant total pressure condition across the contact discontinuity is broken as well. This many seen non-physical, but we note that Eq. (1) or Eq. (3) should be considered as an extra dissipation term when the compressibility of the slow mode is important. In the following, we numerically examine the effects of using these two different formulas in the dissipation term of the HLLD-ec scheme. To minimize other numerical effects, the simulations were first-order accurate in space and time, and the divergence constraint was imposed using a constrained transport method [5] . The test cases below show that the simple modification in Eq. (3) is effective. The present solution is again significantly more robust than the classic HLLD scheme [2] particularly under strong magnetic fields, at the cost of being more diffusive for the slow mode and contact discontinuity. The Dai-Woodward shock tube problem [4]: Figure 1: Results of the Dai-Woodward shock-tube problem, at t = 0 .2. This shock-tube problem involves all 7 MHD waves in the MHD Riemann problem. This 1D problem within x ∈ [−0 .5 , 0.5] has two sets of in

FAQs

What problem is found in the original HLLD-ec scheme?

In the Dai–Woodward-like slow-magnetoacoustic shock test with enhanced longitudinal magnetic field, the original HLLD-ec scheme produces an unphysical oscillation in the density distribution.

How does the proposed fix address the unphysical density oscillation?

The fix changes the intermediate internal-energy formula so that gas pressure is no longer assumed constant between Alfvén waves; instead it is allowed to vary proportionally with intermediate densities, effectively modifying the dissipation behavior.

How does the revised scheme perform compared with the classic HLLD method?

The revised HLLD-ec remains significantly more robust than the classic HLLD scheme under strong magnetic fields, although it is more diffusive for the slow mode and contact discontinuity.

Jul 16, 2026

Upload by Seraphina