Shock Waves

, Volume 28, Issue 2, pp 379–399 | Cite as

Development of a particle method of characteristics (PMOC) for one-dimensional shock waves

  • Y.-H. Hwang
Original Article


In the present study, a particle method of characteristics is put forward to simulate the evolution of one-dimensional shock waves in barotropic gaseous, closed-conduit, open-channel, and two-phase flows. All these flow phenomena can be described with the same set of governing equations. The proposed scheme is established based on the characteristic equations and formulated by assigning the computational particles to move along the characteristic curves. Both the right- and left-running characteristics are traced and represented by their associated computational particles. It inherits the computational merits from the conventional method of characteristics (MOC) and moving particle method, but without their individual deficiencies. In addition, special particles with dual states deduced to the enforcement of the Rankine–Hugoniot relation are deliberately imposed to emulate the shock structure. Numerical tests are carried out by solving some benchmark problems, and the computational results are compared with available analytical solutions. From the derivation procedure and obtained computational results, it is concluded that the proposed PMOC will be a useful tool to replicate one-dimensional shock waves.


Particle method of characteristics (PMOC) One-dimensional shock waves Rankine–Hugoniot relations Shock tube problems 

List of symbols


Coefficient matrix


Channel area in open-channel flow


Flow width in open-channel flow


Convection velocity or flow sound speed


Courant number, \(c{\Delta } x/{\Delta } t\)


Flux function in conservation form


Generalized fluid density function


Flow piezometric head


Total particle number


Initial particle number




Right-side eigenvector matrix




Conserved solution vector


Particle conveying velocity


Primitive solution vector


Fluid velocity


Characteristic variable vector or Riemann invariant vector


Flow characteristic


Horizontal coordinate

Greek symbols

\(\beta \)

Barotropic coefficient

\({\Delta } t\)

Computational time step

\(\delta \)

Initial particle spacing

\(\delta t\)

Incremental time step to derive Rankine–Hugoniot relation

\(\delta x\)

Incremental spatial step to derive Rankine–Hugoniot relation

\(\varepsilon \)

Solution error

\(\eta \)

Flow depth in open-channel flow

\(\varvec{{\varLambda }}\)

Diagonal matrix

\(\lambda \)

Characteristic transmitting velocity

\(\rho \)

Fluid density

\(\sigma \)

Shock speed

\(\phi \)

Flow driving function

\(\varphi \)

Flow characteristic function

\(\psi \)

Reduced flow characteristic function





Exact solution


Open-channel flow


Grid index


Inserted particle




Left neighboring particle


Dividing location for shock tube problems


Criterion for particle insertion spacing


Right neighboring particle


Reference state in two-phase flow




Left side of a shock


Right side of a shock


Reference state for open-channel flow



Previous time step


Fictitious or intermediate states






The author would like to express his gratitude to two anonymous referees for their constructive criticisms and suggestions which were very beneficial in the revision of this article.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Marine EngineeringNational Kaohsiung Marine UniversityKaohsiungTaiwan

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