# Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials

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## Abstract

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## 1. Introduction

## 2. Foundation of NMM

## 3. Construction of Local and Global Approximations

#### 3.1. Local Approximation of Zero Order

#### 3.2. Local Approximation of High Order

#### 3.3. Local Approximation on Singular Patch

## 4. Discrete Equations

#### 4.1. Known Function with Discontinuities

#### 4.2. The Derivative

## 5. Numerical Tests

#### 5.1. Checking the Linear-Dependence Issue and Stability

#### 5.2. The Convergency with More Initial Mathematical Cover

#### 5.3. The Convergency with Higher Order Local Approximation

#### 5.4. 1D Advection Equations

## 6. Conclusions

- NMM can approximate the solution using different local approximation, and the local approximation can be any order.
- The higher order NMM has bigger kernel space with providing better precision.
- NMM can be successfully used to deal with the discontinuity capture, due to the divisible physical patches.
- Although there are discontinuities, the “rank problem” (linear dependence) will not occur when the physical patches are cut. The uniqueness and existence of solution can be verified by numerical examples.
- For dealing with discontinuities, we can cut the physical patches at discontinuities, and add more equations at discontinuities. We can also cover the discontinuity with mathematical patches by using exact solver such as Riemann solver.
- The speed of the discontinuities should be determined by the Ranking condition when discontinuities occur.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NMM | Numerical manifold method |

FEM | finite element methods |

FDM | finite difference methods |

FVM | finite volume methods |

ENO | essentially non-oscillatory behavior |

TVD | Total Variation Diminishing |

MUSCL | Monotone Upstream-centered Schemes for Conservation Law |

LAs | local approximations |

GFEM | generalized finite element methods |

MC | mathematical cover |

PC | physical cover |

PPs | physical patches |

GFEM | generalized finite element methods |

MLS | Moving Least-Squares |

LD | linear dependence |

PU | partition of unity |

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**Figure 4.**Without discontinuities, the rank of the coefficient matrix is always equal to the number of physical patches.

**Figure 5.**When the number of initial mathematical patches is different, as the number of discontinuities increases, the rank of the matrix and the number of PPs also change. The initial number of MPs are $2,16,32$.

**Figure 6.**As the number of cuts increases, the function approximation becomes more and more accurate. When the number of discontinuities is equal to the number of cuts, the function in the figure can be accurately approximated. The discontinuities of the function are at $2.5,5$ and $7.5$.

**Figure 7.**As the number of cuts increases, the number of physical patches is always equal to the rank of the matrix, thus avoiding the problem of linear correlation causing equations that cannot be solved. Using the method in this article, the robustness will be better.

**Figure 8.**With the increase in the number of mathematical patches, the result of using NMM approximation is getting closer and closer to the real solution. Therefore, the convergence of this method can be verified.

**Figure 9.**In the case of the same number of mathematical patches, as the order of local approximation increases, the result of global approximation becomes more and more accurate.

**Figure 10.**(

**a**) The result of using the traditional finite volume method to solve the convection equation at time 0.5. (

**b**) Using the method in this paper to solve the convection equation at time 0.5.

**Table 1.**The convergency of NMM with initial mathematical cover, the error is decreasing when the initial MPs is increasing.

MPs Number | Discontinuities Number | PPs Number | Err | lg(Err) |
---|---|---|---|---|

2 | 3 | 8 | 0.05 | −1.301029996 |

4 | 3 | 10 | 0.352 | −1.453457337 |

8 | 3 | 14 | 0.0139 | −1.8569852 |

16 | 3 | 22 | 0.0034 | −2.468521083 |

32 | 3 | 38 | 8.17 × 10^{−4} | −3.087549428 |

64 | 3 | 70 | 2.01 × 10^{−4} | −3.696587929 |

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**MDPI and ACS Style**

Zeng, Y.; Zheng, H.; Li, C.
Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials. *Appl. Sci.* **2020**, *10*, 9123.
https://doi.org/10.3390/app10249123

**AMA Style**

Zeng Y, Zheng H, Li C.
Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials. *Applied Sciences*. 2020; 10(24):9123.
https://doi.org/10.3390/app10249123

**Chicago/Turabian Style**

Zeng, Yan, Hong Zheng, and Chunguang Li.
2020. "Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials" *Applied Sciences* 10, no. 24: 9123.
https://doi.org/10.3390/app10249123