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**The characteristic-based-split procedure: An efficient and accurate algorithm for fluid problems.**
*(English)*
Zbl 0985.76069

From the summary: We describe an algorithm designed to replace the Taylor-Galerkin (or Lax-Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm is applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective-diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor-Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system, while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible state is reached.

### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76N15 | Gas dynamics (general theory) |

### Keywords:

characteristic-based-split procedure; compressible flow; convective diffusion; characteristic Galerkin procedure; Taylor-Galerkin procedure
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\textit{O. C. Zienkiewicz} et al., Int. J. Numer. Methods Fluids 31, No. 1, 359--392 (1999; Zbl 0985.76069)

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