wiki:u/erica/RoeSolver

Version 46 (modified by Erica Kaminski, 11 years ago) ( diff )

Recap

The Euler equations are a set of non-linear PDE's, which comprise an eigenvalue problem. The eigenvalues for the equations are functions of the solution to the equations themselves. This means that the waves, which propagate with speeds = to their eigenvalues, distort the solution, and the solution distorts them over space and time. Thus the solution to the Riemann problem for the non-linear system does not consist of a closed form expression for the values of pstar and ustar like it does for a linear system of equations. To solve the Euler equations exactly then, we have developed the method of characteristics that describe the propagation of waves outside of the star region. To solve for values of the q-array inside of the star region, we used an iterative scheme and then sampled the solution in the different wave regions. We now are concerned with approximations to this exact solution. The method to be discussed here considers a 'linearized' version of the Euler equations, so analytical methods used for linear, constant coefficient systems of equations can be applied.

The ROE Solver

The Roe solver is an approximation means for the numerical flux of the Godunov method, which is derived through linearizing a hyperbolic system of equations. For instance, the Euler equations in conservative form are written

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which using the chain rule is identical to

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If we assume that

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where, A-hat is a Jacobian matrix of averaged/constant values, we can derive an expression for the numerical flux in terms of 1) wave strengths (alpha), and the 2) eigenvalues (lambda) and 3) right eigenvectors (K) of the 'averaged' Jacobian:

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where

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So the goal is to compute the wave speeds and associated eigenvalues and eigenvectors of the Jacobian matrix. There are 2 methods by which we can do this: 1) The 'Roe' approach, which (non-trivially) constructs an averaged Jacobian directly that must satisfy conditions such as hyperbolicity and conservation, and 2), the newer 'Roe-Pike' approach, which avoids solving for the Jacobian and instead develops algebraic expressions for the sought quantities based on averages of the initial data. It is the 2nd, more widely used approach, that we will explore here.

The Roe-Pike Approach

This approach avoids construction of an averaged Jacobian. Instead the method involves solving for the eigen values and vectors for the original Jacobian, deriving an expression for the wave speeds, and then evaluating these expressions using some vector of averaged scalar quantities, typically W, averaged. So from the equation above of the Euler equations in Jacobian form, we can derive the equations for wave speeds:

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The eigen values and vectors are derived straightforwardly from the Jacobian, using typical methods.

Now, we can either assume the independent variables u, rho, etc. of these functions for alpha, lambda, and K are 1) reference states which we set, or 2) some general averaged versions of the variables, which we have to solve for. Toro goes through the algebraic analysis for the general case for the Euler equations. The results are as follows.

Roe-Pike Approach for Euler Equations

For the x-split, 3-dimensional Euler equations, we have the following eigen values and vectors:

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and

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where u, v, w, are velocities in x, y, z directions, respectively, a is the local sound speed, given by

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H is the enthalpy given by

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E is the total energy per unit volume,

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with

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and e being the specific internal energy, which for ideal gases is,

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Recall that the numerical flux is given by,

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where the tilde's are the solution of the Roe approach, and are given by:

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A Sample Algorithm

Short-comings of the Roe Solver

The code

Results

Discussion

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