Abstract
Our analysis has so far been confined to scalar conservation laws in one dimension. Clearly, the multidimensional case is considerably more important. Luckily enough, the analysis in one dimension can be carried over to higher dimensions by essentially treating each dimension separately. This technique is called dimensional splitting. The final results are very much the natural generalizations one would expect.
The same splitting techniques of dividing complicated differential equations into several simpler parts can in fact be used to handle other problems. These methods are generally called operator splitting methods or fractional steps methods.
Just send me the theorems, then I shall find the proofs.Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200. — Chrysippus told Cleanthes, 3rd century BC Just send me the theorems, then I shall find the proofs. — Chrysippus told Cleanthes, 3rd century BC Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200.
Just send me the theorems, then I shall find the proofs.
— Chrysippus told Cleanthes, 3rd century BC
Lucky guy! Paraphrased from Diogenes Laertius, Lives of Eminent Philosophers, c. a.d. 200.
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Notes
- 1.
If we want a solution for all time, we disregard the last term in (4.4) and integrate t over \([0,\infty)\).
- 2.
We will keep the ratio \(\lambda={\Updelta t}/\Updelta x\) fixed, and thus we index only with \({\Updelta t}\).
- 3.
In several dimensions the CFL number is defined as \(\max_{i}(\left|f_{i}^{\prime}\right|{\Updelta t}/{\Updelta x}_{i})\).
- 4.
Although we have used the parabolic regularization to motivate the appropriate entropy condition, we have constructed the solution of the multidimensional conservation law independtly, and hence it is logically consistent to use the solution of the conservation law in combination with operator splitting to derive the solution of the parabolic problem. A different approach, where we start with a solution of the parabolic equation and subsequently show that in the limit of vanishing viscosity the solution converges to the solution of the conservation law, is discussed in Appendix B.
- 5.
Essentially replacing the operator H used in operator splitting with respect to diffusion by R in the case of a source.
- 6.
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Holden, H., Risebro, N.H. (2015). Multidimensional Scalar Conservation Laws. In: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences, vol 152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47507-2_4
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DOI: https://doi.org/10.1007/978-3-662-47507-2_4
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