Positivity- -preserving Lagrangian schemes for preserving - - PowerPoint PPT Presentation

Positivity Positivity-

• preserving Lagrangian schemes for

preserving Lagrangian schemes for multi multi-

• material compressible flows

material compressible flows

Juan Cheng ( Juan Cheng (成 成 娟 娟) ) Institute of Applied physics and Computational Mathematics cheng_juan@iapcm.ac.cn Joint work with Chi-Wang Shu “Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems”, May 22-26, 2014，Beijing

Outline

• Introduction
• The positivity-preserving HLLC numerical flux for the

Lagrangian method

• The high order positivity-preserving Lagrangian schemes
• Numerical results
• Concluding remarks

• I. Introduction

Inertial Confinement Fusion Astrophysics

Multi-material problems

Underwater explosion

g heavy heavyρ ρ1

1

light lightρ ρ2

2

Methods to Describe Fluid Flow

 Eulerian Method

The fluid flows through a grid fixed in space

 Lagrangian Method

The grid moves with the local fluid velocity

 ALE Method

(Arbitrary Lagrangian-Eulerian )

The grid motion can be chosen arbitrarily

The Lagrangian Method



Can capture the material interface automatically



Can maintain good resolution during large scale compressions/expansions



Is widely used in many fields for multi-material flow simulations

Astrophysics, Inertial Confinement Fusion (ICF), Computational fluid dynamics (CFD), ……

The property of positivity-preserving for the numerical method



As one mathematical aspect of scheme robustness, the positivity-preserving property becomes more and more important for the simulation of fluid flow.



At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may lead to negative density or internal energy.



This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian method.

The positivity-preserving Eulerian method

 low order schemes



Godunov scheme



The modified HLLE scheme



Lax-Friedrichs scheme



HLLC scheme



AUSM+ scheme



Gas-kinetic schemes



Flux vector splitting schemes  high order schemes (Zhang & Shu et al.)



Runge-Kutta discontinuous Galerkin (RKDG) schemes



weighted essentially non-oscillatory (WENO) finite volume schemes



WENO finite difference schemes

Up to now, no positivity-preserving Eulerian schemes have involved multi-material problems.



The Godunov-type Lagrangian scheme based on the modified HLL Riemann solver



C.D. Munz, SIAM Journal on Numerical Analysis, 1994.



The positive and entropic Lagrangian schemes for gas dynamics and MHD.



• F. Bezard and B. Despres, JCP, 1999;



• G. Gallice, Numerische Mathematik, 2003.

The positivity-preserving Lagrangian method

• nly first order accurate.
• nly valid in 1D space.
• impossible to be extended to higher dimensional space due to

the usage of mass coordinate.

The positivity-preserving Lagrangian schemes for multi-material flow

We will discuss the methodology to construct the positivity- preserving Lagrangian schemes. We will propose



a positivity-preserving HLLC approximate Riemann solver for the Lagrangian schemes



a class of positivity-preserving Lagrangian schemes



1st order & high order



1D & 2D



multi-material flow



general equation of state (EOS)

• II. The positivity-preserving

HLLC numerical flux for the Lagrangian method

The compressible Euler equations in Lagrangian formulation

Equation of State (EOS)

The general form of the cell-centered Lagrangian schemes

Numerical flux The first order scheme

the left and right values of the primitive variables on each side of the boundary



1st order:



high order: reconstruction

The HLLC numerical flux for the Lagrangian scheme

the similarity solution along the contact wave

Simplified Riemann fan for HLLC flux

The choice of left and right acoustic wave speeds

• divergence theorem
• G is a convex set
• Jensen’s inequality for integrals

The first order positivity preserving Lagrangian scheme in 1D space

Theorem:

The first order Lagrangian scheme for Euler equations with the general EOS in 1D space is positivity-preserving if the acoustic wavespeeds S- and S+ and the time step restriction are satisfied:

The general form for 2D Lagrangian schemes

HLLC flux for the Lagrangian scheme

Theorem: The 2D first order Lagrangian scheme is positivity- preserving if the acoustic wavespeeds S- and S+ and the time step restriction are satisfied:

The first order positivity-preserving Lagrangian scheme in 2D

1st order scheme

• III. The high order positivity-

preserving Lagrangian schemes

The high order positivity-preserving Lagrangian scheme in 1D space

ENO reconstruction (Euler forward time discretization)

where

1st order scheme:

Theorem:

The 1D high order Lagrangian scheme is positivity- preserving if it uses the above described HLLC flux and satisfies: the time step restriction: the sufficient condition:

The high order positivity-preserving Lagrangian scheme in 1D space

The positivity-preserving limiter for the high order Lagrangian scheme



Firstly, enforce the positivity of density,



Secondly



enforce the positivity of internal energy e for the cells with the ideal EOS or the JWL EOS,



enforce the positivity of for the cells with the stiffened EOS,



Finally This limiter can keep accuracy, conservation and positivity.

The high order time discretization for the Lagrangian scheme

At each Runge-Kutta step, we need to update:

• conserved variables
• vertex velocity
• position of the vertex
• size of the cell

The TVD Runge-Kutta method

The third order TVD Runge-Kutta method

Step 1 Step 2 Step 3

The scheme with the Euler forward time discretization

The high order positivity-preserving Lagrangian scheme in 2D

 ENO reconstruction  Numerical flux



The HLLC flux

 Gaussian integration for the line integral

The Gauss-Lobatto quadrature for the polynomials in cells with general quadrilateral shape

coordinate transformation

The design of 2D high order positivity- preserving Lagrangian scheme

a formal 2D first order positivity-preserving scheme formal 1D first order positivity-preserving schemes

Theorem:

The 2D high order Lagrangian scheme is positivity- preserving if it uses the above described HLLC flux and satisfies: the time step restriction: the sufficient condition:

The high order positivity-preserving Lagrangian scheme in 2D

• IV. Numerical results

 1D case  2D case

All the numerical examples shown here can’t be simulated by the general high order Lagrangian schemes without the positivity-preserving limiter successfully.

• 1. The isentropic smooth problem (accuracy test)

1D Numerical tests

density velocity Internal energy time=0.1

The initial condition :

Errors for the 1st order positivity-preserving Lagrangianscheme

Errors for the 3rd order positivity-preserving Lagrangianscheme

• 2. 123 problem
• contain vacuum

The initial condition :

400 cells time=1.0 density velocity internal energy

Non positivity-preserving & positivity-preserving

density pressure internal energy

• 3. The gas-liquid shock-tube problem

The initial condition :

• multi-material
• strong interfacial contact discontinuity

200 cells time=0.00024 density velocity internal energy

• 4. The spherical underwater explosion

The initial condition :

• multi-material
• general EOS
• large pressure jump

Lagrangian

density pressure

• C. Farhat et al., Journal of Computational Physics, 231 (2012) 6360-6379.

Eulerian

) 1 , 1 ( ) , ( , 1 . 1    v u p 

The initial condition：

Add an isentropic vortex perturbations：

• 1. the vortex problem (accuracy test)

The mean flow:

lowest density： lowest pressure：

2D Numerical Tests

Errors for the 1st order positivity-preserving Lagrangian scheme

Errors for the 2nd order positivity-preserving Lagrangianscheme

• 2. The Sedov blast wave problem

The initial condition (on the domain with cells): The exact solution： A shock at the radius=1 with a peak density of 6 at the time=1.

14

1, 10 0, 1.4 (1,1) 182.09

x y

p u u e  



      ，

1 . 1 1 . 1  30 30

1st order 2nd order Time=1 density pressure grid time=1

• 3. The air-water-air problem

Initial condition:

t=0.007 t=0.003 t=0.0015 density grid 2nd order

t=0.007 t=0.003 t=0.0015 velocity internal energy 2nd order

t=0.007 t=0.003 t=0.0015

velocity internal energy density The cut contour results

• V. Concluding remarks

We have described the general techniques to construct the positivity-preserving Lagrangian schemes for the compressible Euler equations with the general equation of state both in 1D and 2D space.

• a positivity-preserving HLLC approximate Riemann solver for the

Lagrangian scheme.

• a class of first order and high order positivity-preserving Lagrangian

schemes.

Future work:



Generalize the schemes to cylindrical coordinates.



Improve the robustness of the high order positivity-preserving Lagrangian schemes.

Reference



• J. Cheng, C.-W. Shu, Positivity-preserving Lagrangian

scheme for multi-material compressible flow, submitted to Journal of Computational Physics, 257, 143– 168, 2014.