In this paper, we extend the entropy scheme for hyperbolic conservation laws to one-dimensional convection-diffusion equation. The operator splitting method is used to solve the convection-diffusion equation that is divided into conservation and diffusion parts, in which the first-order accurate entropy scheme is applied to solve the conservation part and the second accurate central difference scheme is applied to solve the diffusion part. Numerical tests show that the

In this paper, we consider the convection-diffusion equation:

In [

The significance of the entropy scheme is in methodology. The original Godunov scheme is first-order accurate [

With the entropy scheme designed in this way, it maintains to be local as the original Godunov scheme. Since all the principle and augmented quantities have solid physical meanings and the reconstruction satisfies all the physical algebraic relations among them, the reconstructed solution in each cell physically well simulates the exact solution, even the latter is not smooth in the cell. Important physical properties such as the entropy condition and nonnegativity of mass and pressure are maintained in the scheme. Moreover, the numerical dissipations are quantitatively controlled in that they are used only near discontinuities and extremes of the solution.

In this paper, we mainly introduce the idea, and we choose a kind of convection-diffusion equation in one dimension with the convection part as

In this paper, we follow [

The outline of the paper is as follows: in Section

We consider the following initial value problem for the convection-diffusion equation:

Multiplied by

For simplicity, we use uniform cells with the cell size

In this way, the solution to equation (

As in [

The entropy scheme with the half-step reconstruction is used to solve equation (

A piecewise constant function with a half step is used to reconstruct the solution in each cell:

In order to compute the HS

Solve the initial value problem (IVP) as follows:

For the linear equation, the exact solution to the problem is

Compute

In practice, we compute

We use central difference to approximate the second derivatives and use the Euler forward time discretization for equation (

We use the operator splitting method so that the initial problem (

Solve the conservation part of equation (

Solve the diffusion part of equation (

This gives the final solution

The entropy scheme described in Section

In this section, we use the entropy scheme to compute one-dimensional convection-diffusion equation. In the following, two examples come from [

Consider the following initial value problem:

The exact solution to this problem is

We take

Example

Cells | Order | Order | ||
---|---|---|---|---|

10 | 1.611 | — | 8.543 | — |

20 | 6.794 | 1.245 | 1.244 | 2.779 |

40 | 2.249 | 1.594 | 1.250 | 3.315 |

80 | 5.346 | 2.072 | 7.563 | 4.047 |

160 | 1.388 | 1.945 | 5.398 | 3.808 |

320 | 4.017 | 1.788 | 4.052 | 3.735 |

640 | 9.336 | 2.105 | 2.113 | 4.261 |

Example

Cells | Order | Order | ||
---|---|---|---|---|

10 | 6.221 | — | 6.749 | — |

20 | 2.727 | 1.189 | 1.237 | 2.447 |

40 | 1.081 | 1.334 | 1.563 | 2.984 |

80 | 6.226 | 0.796 | 1.602 | 3.286 |

160 | 1.509 | 2.044 | 7.360 | 4.444 |

320 | 2.582 | 2.547 | 2.784 | 4.724 |

640 | 3.933 | 2.714 | 1.332 | 4.385 |

Consider the following initial value problem.

Equation (

Example

Cells | Order | Order | ||
---|---|---|---|---|

10 | 3.168 | — | 1.629 | — |

20 | 7.198 | 2.137 | 1.045 | 3.962 |

40 | 2.302 | 1.644 | 8.759 | 3.577 |

80 | 5.877 | 1.970 | 5.960 | 3.877 |

160 | 1.561 | 1.912 | 3.916 | 3.927 |

320 | 4.321 | 1.853 | 2.640 | 3.890 |

640 | 1.222 | 1.822 | 2.046 | 3.689 |

In this paper, the entropy scheme is extended to one-dimensional of convection-diffusion equation. We divide the convection-diffusion equation into two parts and use the operator splitting method to solve it. The first-order accurate entropy scheme is applied to solve the conservation part, and the second accurate central difference scheme is applied to solve the diffusion part. We have presented two numerical examples, and the numerical results show that the

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest.

The research was supported by the National Natural Science Foundation of China (no. 11201436) and Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan).