Riemann Solvers

Code snippets follow from Riemann Solvers and Numerical Methods for Fluid Dynamics by Eleuterio F. Toro, where essentials of CFD are discussed in detail.

Linear Advection(ch2 & ch5 & ch13)

Both smooth and discontinous initial velocity profile are examined.
The exact solution is quite trival, just trace back along the characteristic line.
Different schemes are employed for comparison:

• CIR
• Lax-Friedrichs
• Lax-Wendroff
• Warming-Beam
• Godunov
• WAF

Usage:

• Compile: g++ smooth.cc -std=c++11 -o advection.out or g++ discontinuous.cc -std=c++11 -o advection.out
• Execute: ./advection.out
• Plot: python3 animate.py data1.txt data2.txt
  (data1.txt and data2.txt are the 2 cases you want to compare)

Invisid Burgers Equation(ch2 & ch5)

Only the discontinous initial velocity profile is examined.
Analytically, the exact solution is either a shock wave or a rarefaction wave. Different schemes are employed for comparison:

• CIR
• Lax-Friedrichs
• Lax-Wendroff
• Warming-Beam
• Godunov

Usage:

• Compile: g++ main.cc -o -std=c++11 burgers.out
• Execute: ./burgers.out
• Plot: python3 animate_single.py or python3 animate_all.py

Euler Equation

1-D Euler equation(ch3) with ideal gases.

Exact solution(ch4)

The general solution of the exact solution follows the 3-wave pattern, where the contact must lies in between, shock and rarefaction waves stay at left or right.
The exact solution can be calculate numerically, where a iterative procedure is necessary for solving the pressure. The exact solution requies much computational effort and this is why approximate riemann solvers are studied extensively back in 1980s. Especially, vacuum condition is considered for completeness.

Usage:

• Compile: g++ main.cc -o Exact.out
• Execute: ./Exact.out < inp.dat
• Plot: python3 animate.py

Godunov's Method(ch6)

The essential ingredient of Godunov's method is to solve Riemann Problem locally, and the keypoint in numerical parctice is to identify all of the 10 possible wave patterns so that the inter-cell flux can be calculated properly.

Usage:

• Compile: g++ main.cc -std=c++11 -o Godunov.out
• Execute: ./Godunov.out < inp.dat
• Plot: python3 animate.py

Lax-Friedrichs(ch5 & ch6)

Follow the guide to rewrite the differential scheme into conservative form.
It's interesting to notice that Lax-Friedrichs scheme is identical to the Riemann Solution averaged at the half of each time step. The key part in parctice is the Lax-Friedrichs inter-cell flux, see (5.77).

Usage:

• Compile: g++ main.cc -std=c++11 -o Lax.out
• Execute: ./Lax.out < inp.dat
• Plot: python3 animate.py

Richtmyer(ch5 & ch6)

Again, rewrite the differential scheme into its conservative form.
Lax-Wendroff is of 2nd-order accuracy in space and time.
The Richtmyer version(2-step Lax-Wendroff), introduces an intermediate step where a tempory conservative variable is computed, and the inter-cell flux is calculated later on based on this tempory conservative variable.

Usage:

• Compile: g++ main.cc -std=c++11 -o Richtmyer.out
• Execute: ./Richtmyer.out < inp.dat
• Plot: python3 animate.py

FVS(ch8)

Here, the inter-cell flux is not calculatd directly from the exact solution of Riemann Problem. Instead, the flux at each point is splitted into 2 parts: upstream traveling and downstream traveling, then, for each inter-cell, the flux is seen as the sum of the upstream traveling part from the left point and the downstream traveling part from the right point.
3 typical splitting techniques are introduced.

Steger-Warming

Usage:

• Compile: g++ main.cc -std=c++11 -o SW.out
• Execute: ./SW.out < inp.dat
• Plot: python3 animate.py

van Leer

Usage:

• Compile: g++ main.cc -std=c++11 -o vL.out
• Execute: ./vL.out < inp.dat
• Plot: python3 animate.py

AUSM

Usage:

• Compile: g++ main.cc -std=c++11 -o AUSM.out
• Execute: ./AUSM.out < inp.dat
• Plot: python3 animate.py

Approximate-state(ch9)

Godunov's method in conjunction with exact Riemann solution spends lots of computational effort on solving pressure in star region iteratively. One kind of approximate Riemann solution seeks to approximate the state of the wave patterns so that unkonwn values at star region can be solved immediately.
5 typical approximate-state Riemann Solvers are implemented.

PVRS

Usage:

• Compile: g++ main.cc -std=c++11 -o PVRS.out
• Execute: ./PVRS.out < inp.dat
• Plot: python3 animate.py

TRRS

Usage:

• Compile: g++ main.cc -std=c++11 -o TRRS.out
• Execute: ./TRRS.out < inp.dat
• Plot: python3 animate.py

TSRS

Usage:

• Compile: g++ main.cc -std=c++11 -o TSRS.out
• Execute: ./TSRS.out < inp.dat
• Plot: python3 animate.py

AIRS

Usage:

• Compile: g++ main.cc -std=c++11 -o AIRS.out
• Execute: ./AIRS.out < inp.dat
• Plot: python3 animate.py

ANRS

Usage:

• Compile: g++ main.cc -std=c++11 -o ANRS.out
• Execute: ./ANRS.out < inp.dat
• Plot: python3 animate.py

HLL(ch10)

By esitmate the 2 fastest wave spreading speed, the averaged flow variable in between are constant!

Direct estimation under Roe-average

The wave spreading speed at left and right front are esitmated according to corresponding eigenvalues. The key part in practice is the Roe-averaged eigenvalues.

Usage:

• Compile: g++ direct.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

Pressure-based estimation

Approximate the pressure at star region, and then estimate the speed at wave front from the exact solution of Riemann Problem.

Usage:

• Compile: g++ pressure_based.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

Rusanov

Only the maximun wave spreading speed is approximated according to eigenvalues, this is a much simpler choice for the fastest signal velocities.

Usage:

• Compile: g++ rusanov.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

HLLC(ch10)

An internal wave is added compared with HLL so that contact is better resolved, and there're 2 ways to estimate the signal speed, namely the Direct estimation under Roe-average and the Pressure-based estimation

Usage:

• Compile: g++ direct.cc -std=c++11 -o Euler.out or g++ pressure_based.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

Roe(ch11)

Instead of estimating the signal prompting speed, another approach seeks to approximate the Jacobian matrix with known left and right states such that 3 essential properties(Hyperbolicity, Consistency and Conservation across discontinuities) are satisfied. An ingenious way to construct such a linearized Jacobian matrix is given by Roe, where the famous parameter vector consists the square root of density is introduced. In practice, the inter-cell flux is computed from the wave strength, eigenvalues and eigenvectors explicitly.

Usage:

• Compile: g++ main.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

Osher(ch12)

The key point of Osher scheme, from my point of view, is the integral in phase space, where integration path should be choosen carefully. An instant choice of integration path is referenced to corresponding eigenvalue, where the path is tangential to the corresponding eigenvector! In this way, the determination of intersections and sonic points are required, and they are computed from Generalised Riemann Invariants, where 2 different ordering can be applied. The physical ordering is implemented in my code.

Usage:

• Compile: g++ main.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

MUSCL(ch14)

MUSCL scheme is famous for its novel idea to produce high-order schemes by reconstructing primitive variables. In practice, slope vector at each point is calculated firstly, then the boundary value of each cell is extrapolated with corresponding slope vector. Extrapolated values at boundaries are evolved half of current time-setp from the approximation of governing equations afterwards. Finally, these evolved values are treated as piecewise constant, and are used as the intial profile of Riemann Problem. Once the Riemann problem is solved, the intercell flux can be calculated.

Basic

Basic version of the MUSCL-Hancock Scheme, no limiter applied to the slope vector.
Only the first test case can be used for reference.

Usage:

• Compile: g++ basic.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py

TVD

The TVD version of the MUSCL-Hancock Scheme, 4 limiter functions are provided(SUPERBEE, MINBEE, VANLEER, VANALBDA). Apply the desired limiter function by modifying the source code ranging from line 723-726.
Only the first test case can be used for reference.

Usage:

• Compile: g++ tvd.cc -std=c++11 -o Euler.out
• Execute: ./Euler.out < inp.dat
• Plot: python3 animate.py