Sinu changes may be made befo", publication. SS, or senice by trade 1WDt, tndtmark. TIw "iewls and opinioas 01 I. Republic of China and Phillip Colena Lawrence Livermore National Laboratory PO Box Livermore, California July, Abstract We present a hybrid front tracking I conservative finite difference method for computing discontinuous solutions to systems of hyperbolic conservation laws.
In this method, the tracked front is allowed to move through the finite difference mesh. The coupling to the finite difference method is done by per- forming a finite volume differencing in the irregular cells formed by the intersection of the finite difference grid wtih the regions on either side of the tracked front. Finally, the overall algorithm is designed to make minimum use of the global geometty of the tracked front; in particular, it can be coupled to a volume-of-fluid description of the tracked front.
Numerical results using this method. The solutions computed are for the problem of self-similar shock reflection from an oblique surface, with the incident shock and the rigid shock tube wall treated as tracked fronts. One is front capturing, where the discontinuities are represented as steep gradients spread over a small number of finite difference cells. The discrete divergence form of the difference equations and the addition of suitable dissipation operators are sufficient to insure that the solution converges to a weak solution of the conserva- tion laws satisfying appropriate entropy conditions.
The other approach is front tracking. Each of the approaches has its advantages and disadvantages. The principal advantage of front capturing is its relative simplicity and its generality. Although a finite difference algo- rithm can, in general, be quite complicated, it is lhe same at all cells. In spite of this simplicity. In contrast, tracking methods must contain mechanisms for predicting the formation and propaga- tion of discontinuities. Although considerable progress has been made in providing these mechanisms .
The accuracy with which strongly nonlinear discontinuities produced by sys- tems of conservation laws are represented in a capturing algorithm depends on the establish- ment of a discrete traveling wave that represents the discontinuity the existence of which in t tum depends strongly on the regularity of the finite difference mesh. For example. These problems disappear if the strong shock is tracked. An exam- ple of the latter are reaction fronts in gas dynamics. In that case, a cap- turing calculation on a mesh which resolves only the hydrodynamic length scales fails, since the numerical viscosity in the truncation errors in the method would vastly exceed the physical values, leading to incorrect wave speeds.
This can also be the case even when the jump rela- tions are independent of the diffusion coefficients, as long as the latter are sufficiently small, as is the case with detonation waves . One approach is to use an adaptive grid to resolve the diffusive length scales by clustering mesh points at the front.
However, if one does not need to resolve the details of the diffusive length scales on the multidimensional finite difference grid. In view of the above considerations. Such an idea is not new for compressible flow cal- culations. Indeed, the earliest numerical methods for gas dynamics were Lagrangian methods ,, which automatically track contact discontinuities, while capturing shocks.
The main disadvantage of these methods is that the mesh on either side of the tracked wave loses smootimess if there are large defonnations of the tracked front. This can lead to a loss of stability and accuracy due to the distortions in the mesh, even for those modes which are only weakly coupled to the tracked discontinuity. An alternative to the tracking method described above is to allow the tracked discontinuity to move freely across a regular finite difference mesh.
This type of tracking has been used extensively in calculations where all the discontinuities are tracked and the coupling of the tracked wave to the interior difference algorithm does not preserve conservation fonn ,,[20 ,,. However, once the requirement of global conservation is imposed, the difficulty occurs that the tracked front generally divides cells into two pieces, one of which can be arbitrarily small.
The solution to this difficulty bas been to locally rear- range the mesh geometry in the neighborhood of the front, by merging the small cell fragments with larger ones, for example. In this paper, we will follow the latter approach of allowing the tracked front to move through the mesh. The main new element in this work. In the previous work cited above, the CFL restrictions were circumvented by modifying the grid geometry in order to enlarge the range of influence of small control volumes adjacent to the tracked front. In the present work. Such representations have proved useful in modeling unstable fronts such as material interfaces ,.
Front Tracking in One Dimension We want to compute numerically solutions to the systems of conservation laws of the form U x. OUf discrete approximation to the solution of 2. In addition. The combined algorithm should be conservative overall, i. FR' Away from the tracked front. In the neigh- borhood of the front.
In order to calculate the trajectory of the discontinuity of the p th wave family over the time step. The only two pieces of data we require from the Riemann problem are s. To update the cells containing the tracked front. We consider first the case jA! There is a third approach which has been used in the special case of a tracked material interface in gas dynamics.
The algorithm which we describe here is in a generalization of the latter approach to the case of a general tracked fronL In order to calculate Ulj,. We do so by distributing these flux differences into nearby t cells into which they can be absorbed without loss of stability. This redistribution is done based on a decomposition of the flux differences in terms of right eigenvectors, so that the various components of s oMs are distributed to where they are being propagated in the sense of characteristics, although possibly sooner than expected.
It is not difficult to check that 2. There are a number of remarks to be made about the algOrithnl described above. The first is that. More generally.
Front Tracking for Hyperbolic Conservation Laws | Helge Holden | Springer
Thus the algorithm redistributes flux differences representing the effect of the smooth wave interacting with the tracked discontinuity. That is. Where the discrete initial data is obtained by taking appropriate averages of the exact initial data over the discrete mesh. JJ], comparing the exact and approximate fluxes at the endpoints. Notice that this argument holds independent of whether the discon- tinuity crosses one of the endpoints. In contrast, there is no analogous estimate known for the exact and approximate fluxes for any shock capturing method for systems of equations, i.
JJ, depending only on their ratio. In the case where the tracked front is a linearly degenerate discontinuity, the redistribution algo- rithm given above reproduces in the discretization the qualitative behavior of the solution to the differential equation that infonnation carried by the pth characteristic family does not cross the discontinuity. If the pth wave is genuinely nonlinear, then the increments 8MtJl could, in prin- ciple, be placed on either side of the discontinuity since in any case they would eventually be t swept up by discontinuity due to the convergence of the characteristics of that family.
This fact leaves one free to redistribute the fluxes based on other considerations. If the shock is propagating into a unifonn low pressure ambient. We assume this system to be hyperbolic in the sense that the system 3. We will use a conservative finite difference algorithm for solving 3. Clearly, such a representation is not unique. We require the algorithm obtained either by set- ling in 3. D,-U arguments to zero. We assume that our tracked front divides our computational domain into two components labeled 1 and 2. At time '. If Al'i is zero, then there is no value assigned to uti.
We then wish to calculate the updated values of the areas ArjIJ. In this sec- tion. J is mown, and give the algorithm for calculating Vr;1,'. In panicu- lar, we will assume that. I are given, in addition to niJt a value for the unit normal to the front, pointing from 2 to 1, as well as JiJ , an estimate of the velocity of the front in the direction of n; J.
J are nonzero for one of the time levels. Given this information, our strategy for calculating UrjlJ will be to reconstruct in each mixed cell a local approximation to the geometry in space-time of the tracked front; to use this information, along with the difference algorithm 3. Given AtJ. In the case that Arj. In the case when one of the A's is zero, then the speed :f is required to determine So. J listed in clockwise order.
We also define The calculation of A If J,AI is a straightforward exercise in trigonometry, and will be sketched in an appendix. However, we do note here the consistency requirements AP. We also mention that AI is easily calculated from the A' 's and A's using the divergence theorem. We now can define our finite difference fluxes. First, if hiJ is a variable defined on the grid.
We define 3. AI and FI are taken to be nonzero only for those cells which contain nonzero volumes of both regions at either the old or new times. As was the case in one dimension. In A ",I ] order to have conservation. N are the linearized right eigenvectors of the system 3. J ' then we can define, analogous to 2. Gas Dynamics in Two Dimensions The fonnulation of the conservative tracking algorithm given in the previous two sec- tiom. The first way is in the time step control. If one considers the special case of a tracked front coinciding with one of the mesh lines.
In general. It is not difficult to construct examples of systems of equa- tions for which the above algorithm generates oscillations for time steps violating the CFL time step limit for the first order Godunov method. The second difficulty is that of a lack of an obvious choice of redisttibution algorithm for the component in the characteristic decomposi- tion of the flux difference corresponding to the same family of the tracked wave in the case where the wave is genuinely nonlinear.
This is a particular problem in two dimensions, where there is a component of the flux difference which is purely an artifact of the impossibility of representing exactly in a numerical calculation the spatial structure of a curved discontinuity. One needs to identify that component of the flux difference, and decide on which side of the tracked front to distribute it Rather than attempting to give general so1utions to these difficulties. For the time step problem, we will intertwine the predictor- corrector algorithm with the tracking algorithm in such a way so that.
We will also identify the com- ponent of the flux difference corresponding to the errors in the representation of a curved front. We wish to solve the equations of gas dynamics in conservation fonn in Cartesian geometry, i. To simplify the exposition, we will assume that the equa- tion of state is that of a polytropic gas, i.
The exten- sion to a general convex equation of state can be canied out using the techniques in . The difference algorithm which we will couple to our front tracking method is a second order extension of Godunov's method. We review this method briefly here; for further details.
The extension to the other three cases are straightforward. First, we calculated the effect of spatial derivative in the direction nonnal to the j. The second step of the predictor step estimates the effect on the extrapolated states at the cell edge of spatial gradients in the direction parallel to that cell edge.
It is given by 4. We also assume that the local geometry of the front has been calculated as in the previous section. In panicular. The values u. Another consequence of 4. Thus, by using 4. The left and right states are given by suitable modifications of the predictor step 4. We first calculate '" I u.. It is possible because of 3. For the postshock state, there is an additional con- tribution to the predictor step due to the ftux through the tracked wave: 4.
We do this because we only want contributions due to solution gradients, as opposed to those due to the nonrectangular finite volume differencing near the front; in panicu- lar, if the solution consists of a straight shock separated by two constant states, the predictor step yields a zero increment.
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The particular form is motivated by the extension of the second order Godunov method to general quadrilateral grids discussed in . We must also calculate a predictor step for the tracked front. Since the flux depends only on the Slate in front of the shock. In the case of the tracked wave coinciding with one of the cell edges, this algorithm reduces to the second part 4.
Since this corresponds to a hybridization of two schemes having the same time step restrictions, i. Also, the effects of incompatibilities in the representation of the shock geometry are effectively assigned to OM:! Numerical Results We use as a test problem the problem of self-similar reflection of a shock by an oblique surface . The initial data consists of a planar shock moving into a unifonn fluid with pres- sure Po. The surface in back of the incident shock parallel to its direction of propaga- tion is also a reflecting surface.
The solution to this problem for positive times consists of some reflected wave propagating into the unifonn postshock medium, with a combination of smooth and discontinuous waves contained in region bounded by the incident shock, the reflected wave, and the solid wall boundaries. NonnallYt one expects the solution to be self- similar, i.
In that case. Mach number Ms, the ramp angle a.
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- Front Tracking for Hyperbolic Conservation Laws!
We use the hybrid tracking and capturing algorithm described in the previous section to calculate solutions to this problem. We track the incident shock, while capturing all the waves found behind it. Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology.
The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. From the reviews of the first edition: "It is already one of the few best digests on this topic.
The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style. Serre, MathSciNet "I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course.