positivity-preserving lagrangian schemes for multi...

PositivityPositivity--preserving Lagrangian schemes for preserving Lagrangian schemes for multimulti--material compressible flowsmaterial compressible flows

Juan Cheng (Juan Cheng (成成娟娟))Institute of Applied physics and Computational Mathematics

[email protected]

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

heavyheavyρρ11

lightlightρρ22

Methods to Describe Fluid Flow

Eulerian MethodThe fluid flows through a grid fixed in space

Lagrangian MethodThe 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

simulationsAstrophysics, 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

only first order accurate. only 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


400 cellstime=1.0

density velocity internal energy

Non positivity-preserving & positivity-preserving

density pressure internal energy

3. The gas-liquid shock-tube problem


• multi-material• strong interfacial contact discontinuity

200 cells

time=0.00024

density velocity internal energy

4. The spherical underwater explosion


• 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 vup

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.

141, 10 0, 1.4

(1,1) 182.09x yp u u

e

，

1.11.1 3030

1st order

2nd order

Time=1

density pressuregrid

time=1

3. The air-water-air problemInitial condition:

t=0.007t=0.003t=0.0015

density

grid

2nd order

t=0.007t=0.003t=0.0015

velocity

internalenergy

2nd order

t=0.007t=0.003t=0.0015

velocity

internalenergy

density

The cut contour results

V. Concluding remarksWe 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.

positivity-preserving lagrangian schemes for multi...

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