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Riemann Simulation Tab

The Riemann Simulation Tab contains commands and settings for simulating Riemann problems using a finite difference scheme.

Riemann Simulation Left State

Select the left state of Riemann Problem. This state can be selected by clicking on the phase space panel or selecting text input and typing the coordinates at main panel

Riemann Simulation Right State

Select the right state of Riemann Problem. This state can be selected by clicking on the phase space panel or selecting text input and typing the coordinates at main panel

Riemann Simulation

Start the simulation. The simulation result will be presented in the phase space panel and in other panels whose axes will be chosen in Riemann Simulation Config...

Comparing Finite Difference with Riemann solutions

If left and right states of a Riemann simulation are the same as for the corresponding Riemann solution, one can obtain direct comparison between the two solutions. We show four methods for using input/output capabilities of Eli to enable equality of states.

Method 1

Let us choose the left state \(U_L\) for the Riemann problem. First we set the left state of the Riemann simulation to be \(U_L\). The simplest way of doing this is by opening the Riemann Sim. tab, clicking on the Riemann Sim. Left State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the left state of the Riemann solution to be \(U_L\). To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Left Wave Curve button in the Wave Curves tab. ELI draws the left wave curve from \(U_L\).

Let us choose the right state \(U_R\) for the Riemann problem. We now set the right state of the Riemann simulation to be \(U_R\). We do so by opening the Riemann Sim. tab, clicking on the Riemann Sim. Right State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the right state of the Riemann solution to be \(U_R\). To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Right Wave Curve button in the Wave Curves tab. ELI draws the right wave curve from \(U_R\).

The left and right states are now ready to be used by the Riemann simulation and by the ELI solution. To perform the Riemann simulation, we open the Riemann Sim. Tab and click on the Riemann Simulation button. ELI computes the solution and displays the animated solution in two solution profile windows. The solution is also displayed in several other windows. We are ready to compute the ELI solution. We open the Wave Curves Tab, click on the Riemann Problem button and click near the intersection of the slow and fast wave curves. ELI verifies geometrical compatibility of the slow and fast wave groups

Method 2

Let us choose the left state U_L for the Riemann problem. First we set the left state of the Riemann simulation to be U_L. The simplest way of doing this is by opening the Riemann Sim. Tab, clicking on the Riemann Sim. Left State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the left state of the Riemann solution to be U_L. To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Left Wave Curve button in the Wave Curves tab. ELI draws the left wave curve from U_L.

Let us choose the right state U_R for the Riemann problem. We now set the right state of the Riemann simulation to be U_R. We do so by opening the Riemann Sim. Tab, clicking on the Riemann Sim. Right State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the right state of the Riemann solution to be U_R. To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Right Wave Curve button in the Wave Curves tab. ELI draws the right wave curve from U_R.

The left and right states are now ready to be used by the Riemann simulation and by the ELI solution. To perform the Riemann simulation, we open the Riemann Sim. Tab and click on the Riemann Simulation button. ELI computes the solution and displays the animated solution in two solution profile windows. The solution is also displayed in several other windows. We are ready to compute the ELI solution. We open the Wave Curves Tab, click on the Riemann Problem button and click near the intersection of the slow and fast wave curves. ELI verifies geometrical compatibility of the slow and fast wave groups and displays the Riemann solution in its full glory, if it exists.

Method 3

Let us choose the left state U_L for the Riemann problem. First we set the left state of the Riemann simulation to be U_L. The simplest way of doing this is by opening the Riemann Sim. Tab, clicking on the Riemann Sim. Left State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the left state of the Riemann solution to be U_L. To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Left Wave Curve button in the Wave Curves tab. ELI draws the left wave curve from U_L.

Let us choose the right state U_R for the Riemann problem. We now set the right state of the Riemann simulation to be U_R. We do so by opening the Riemann Sim. Tab, clicking on the Riemann Sim. Right State button and then clicking the mouse on the chosen state in the Phase Space window. ELI responds by marking the chosen state with a cross, printing its coordinates and filling the state coordinates in the Main tab. The next stage consists in setting the right state of the Riemann solution to be U_R. To do so we select the Input Mode to be TextOnce in the Main tab and then click on the Right Wave Curve button in the Wave Curves tab. ELI draws the right wave curve from U_R.

The left and right states are now ready to be used by the Riemann simulation and by the ELI solution. To perform the Riemann simulation, we open the Riemann Sim. Tab and click on the Riemann Simulation button. ELI computes the solution and displays the animated solution in two solution profile windows. The solution is also displayed in several other windows. We are ready to compute the ELI solution. We open the Wave Curves Tab, click on the Riemann Problem button and click near the intersection of the slow and fast wave curves. ELI verifies geometrical compatibility of the slow and fast wave groups

Tolerances

  • a_tol

  • b_ tol

  • c_ tol

Riemann Simulation Config...

PDE Terms

Upwind Flux

when the flux is approximated via upwind scheme

Diffusion Term

Choose the diffusion matrix configurated in Model -> Viscosity Params Settings ... with the parameters coefPwo , coefPgo, Adjugate Choice and Capillarity Form. Doctoral thesis Luis Lozano

Constant Diffusion

Choose the constant diffusion matrix (b00 b01 b10 b11) with the values chosen in Model -> Viscosity Params Settings ...

Flux Term Implicit

Configuration of the finite differences scheme:

  • 0 for fully explicit
  • 1 for fully implicit terms.
  • 0.5, then becomes the Crank-Nicolson . Any other value, if it makes sense, will also be accepted

Notice that when a variable is set to 0, the corresponding term does not need to have derivatives implemented inside jet function. However, for every other value, the derivatives must be provided.

Space Mesh Points

Will be used to define number of mesh points

Final Simulation Time

Final time of the simulation

Speed Domain Percentage

Controls the x axis length in profile windows panels .

Numerical Scheme

Choice of numerical schemes used in numerica simulation

CFL Cond. Percentage

Fraction of Courant-Friedricks-Lewy time step size to be actually used

Adaptative CFL

The time step above is chosen adaptatively by algorithm

Adaptive CFL Frequency

Flux Term Implicity

Flux term implicitly or Explictly according to the value of this parameter

  • 0 Explict
  • 1 Implicit
  • 0.5 Crank-Nicolson

Diffusion Term Implicity

Diffusion term implicitly or Explictly according to the value of this parameter

  • 0 Explict
  • 1 Implicit
  • 0.5 Crank-Nicolson