# Shock Wave: bvpT24

shock wave problem: bvpT24 | |
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Contributor: | testset of J.R. Cash |

Discipline: | fluid dynamics |

Accession: | 2013 |

**Short description:**

The problem describes a shock wave in a one dimension nozzle flow. The steady state Navier-Stokes equations generate a second order differential equations that is reduced to a first order system of 2 equations.

**Applicable solvers:**

all the solvers supported by the Test Set.

**Plots of the solution**** <- **click to generate the plots of the solution and the textual output

**Mathematical description:**

Consider a shock wave in a one dimension nozzle flow. The steady state Navier-Stokes equations give

where t is the normalized downstream distance from the throat, z is the normalized velocity, A(t) is the area of the nozzle at t , with

We write this problem in first order form by defining and , yielding a system of differential equations of the form

where

with

The boundary conditions are obtained from

Given its simple appearence, the BVP turns out to be a surprisingly difficult numerically. An shock develops, whose location depends on .

Singular-perturbation-type problems usually require a continuation method to solve them .For this BVP, however, many steps need to be taken.

**Download:**

- Fortran code: bvpT24.f
- matlab code: bvpT24.m
- R code: first order: bvpT24.R, high order: bvpT24_ho.R