The degree of a complete polynomial

## Abstract

In this chapter, a new efficient high-order finite volume method for 3D elastic modelling on unstructured meshes is developed. The stencil for the high-order polynomial reconstruction is generated by subdividing the relative coarse tetrahedrons. The reconstruction on the stencil is performed by using cell-averaged quantities represented by the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, the new method has a very local property like the discontinuous Galerkin method. Furthermore, it can be written as an inner-split computational scheme which is beneficial to reducing computational amount. The reconstruction matrix is invertible and remains unchanged for all tetrahedrons, and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. The high-order accuracy in time is obtained by the Runge-Kutta method. Numerical computations including a 3D real model with complex topography demonstrate the effectiveness and good adaptability to complex topography.

### Keywords

- numerical solutions
- computational seismology
- 3D elastic wave
- wave propagation
- high-order finite volume method
- unstructured meshes

## 1. Introduction

Wave propagation based on wave equations has important applications in geophysics. It is usually used as a powerful tool to detect the structures of reservoir. Thus solving wave equations efficiently and accurately is always an important research topic. There are several types of numerical methods to solve wave equations, for example, the finite difference (FD) method [1, 2], the pseudo-spectral (PS) method [3, 4], the finite element (FE) method [5, 6, 7, 8, 9], the spectral element (SE) method [10, 11, 12, 13, 14], the discontinuous Galerkin (DG) method [15, 16, 17, 18], and the finite volume (FV) method [19, 20, 21, 22]. Each numerical method has its own inherent advantages and disadvantages. For example, the FD method is efficient and relatively easy to implement, but the inherent restriction of using regular meshes limits its application to complex topography. The FE method has good adaptability to complex topography, but it has huge computational cost. In this chapter, the FV method is the key consideration.

In order to simulate wave propagation on unstructured meshes efficiently, the FV method is a good choice due to its high computational efficiency and good adaptability to complex geometry. In this chapter an efficient FV method for 3D elastic wave simulation on unstructured meshes is developed. It incorporates some nice features from the DG and FV methods [15, 16, 17, 19, 20, 23] and the spectral FV (SFV) method [24, 25, 26]. In our method, the computational domain is first meshed with relative coarse tetrahedral elements in 3D or triangle elements in 2D. Then, each element is further divided as a collection of finer subelements to form a stencil. The high-order polynomial reconstruction is performed on this stencil by using local cell-averaged values on the finer elements. The resulting reconstruction matrix on all coarse elements remains unchanged, and it can be pre-computed before time evolution. Moreover, the method can be written as an inner-split computational scheme. These two advantages of our method are very beneficial to enhancing the parallelization and reducing computational cost.

The rest of this chapter is organized as follows. In Section 2, the theory is described in detail. In Section 3, numerical results are given to illustrate the effectiveness of our method. Finally, the conclusion is given in Section 4.

## 2. Theory

### 2.1 The governing equation

The three-dimensional (3D) elastic wave equation with external sources in velocity-stress formulation can be written as the following system [1, 15]:

where *é* coefficients and

where

The propagation velocities of the elastic waves are determined by the eigenvalues

where

are the velocities of the compression (

### 2.2 The generation of a stencil

Suppose that the 3D computational domain

In practical computations, the integrals in the FV scheme on physical tetrahedral element

and its corresponding inverse transformation by

The detailed expressions of the transformations (6) and (7) will be given in Section 2.5.

Inside each

where

In order to construct a high-order polynomial, we need to choose a stencil. Traditionally, the elements being adjacent to the element

Let

In our algorithm, we guarantee

1 | 2 | 3 | 4 | |
---|---|---|---|---|

4 | 10 | 20 | 35 | |

2 | 3 | 3 | 4 | |

8 | 27 | 27 | 64 |

### 2.3 The high-order polynomial reconstruction

The high-order polynomial is reconstructed in each element

where

to reconstruct a high-order polynomial, where

To solve the reconstruction problem, inspired by the DG method [15, 16, 17, 23, 28, 29], we use hierarchical orthogonal basis functions. The basis functions

Transforming equation (12) in the physical coordinate system

where

The integration in Eq. (14) over

where

and

We need at least

From the orthogonality of basis functions and the property of Eq. (13), we remark that Eq. (15) is subject to the following constraint condition [27]:

With the constraint, Eq. (15) is solved by the Lagrange multiplier method [19, 20, 27]. And the system can be written as

where

The coefficient matrix on the left-hand side of Eq. (19) is the so-called reconstruction matrix [19, 20].

### 2.4 The spatial discrete formulation

We now derive the semi-discrete finite volume scheme based on Eqs. (2) and (8). Integrating over each subelement

Using Eq. (8) and integration by parts yield

where

where

where

And

where

Inserting Eqs. (23) into (22) and rewriting the result into a splitting form of easy computation in the reference system

with

and

where

where

1 | 2 | 3 | |
---|---|---|---|

### 2.5 The time discretization

Equation (27) is in fact a semi-discrete ordinary differential equation (ODE) system. In order to solve it formally, we denote the spatial semi-discrete part in Eq. (27) by a linear operator

Traditionally, the classic fourth-order explicit RK (ERK) method

can be applied to advance

As we can see the LSERK only requires one additional storage level, while ERK has four. The coefficients required in Eq. (34) are listed in Table 3 [30].

1 | 0 | 0.1496590219992291 | 0 |

2 | −0.4178904744998519 | 0.3792103129996273 | 0.1496590219992291 |

3 | −1.1921516946426769 | 0.8229550293869817 | 0.3704009573642048 |

4 | −1.6977846924715279 | 0.6994504559491221 | 0.6222557631344432 |

5 | −1.5141834442571558 | 0.1530572479681520 | 0.9582821306746903 |

As to the stability condition, it is controlled by the Courant-Friedrichs-Lewy (CFL) condition [15, 19];

where

The absorbing boundary conditions (ABCs) in computations are required as the computational domain is finite. There are two typical ABCs to be adopted here. One is flux type ABCs [16, 19]. That is to say, the following numerical flux in Eq. (23) at all tetrahedral faces that coincide with domain boundary

which allows only for outgoing waves and is equivalent to the first order ABCs. Though the absorbing effects of this method vary the angles of incidence, it is still effective in many cases [19]. The advantage of this type ABCs is that it merged into the FVM framework naturally and there is almost no additional computational cost. Another type is the perfectly matched layer (PML) technique originally developed by [31], which is very popular in recent more 10 years.

### 2.6 Coordinate transformation

The transformation between different coordinate systems is frequently used. For ease of reading, we present the formulations here. Let

then the transformation from

where

Note that

The coordinate transformation from the second reference coordinate

then the transform from

which is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the subelement

## 3. Numerical computations

In this section we give three numerical examples to illustrate the performance of the developed method above. The convergence test of the proposed method can be found in [27]. Though the method is developed for the 3D case, it can be simplified to 2D without essential difficulty. The principle is the same. The first example is a test for a 2D model with uneven topography. The other two examples are for two 3D models.

Example 1. The first example is a two-layered model with the inclined interface shown in Figure 3a. The range of the model is

where

is applied, where

Example 2. The second example is a cuboid model. The physical size of the model is

It is applied to the

Example 3. The third example is a real geological model in China. As shown in Figure 8a, it has a very complex topography. The physical scope of the model is

## 4. Conclusions

A new efficient high-order finite volume method for the 3D elastic wave simulation on unstructured meshes has been developed. It combines the advantages of the DG method and the traditional FV method. It adapts irregular topography very well. The reconstruction stencil is generated by refining each coarse tetrahedron which can be implemented effectively for all tetrahedrons whether they are internal or boundary elements. The hierarchical orthogonal basis functions are exploited to perform the high-order polynomial reconstruction on the stencil. The resulting reconstruction matrix remains unchanged for all tetrahedrons and can be pre-computed and stored before time evolution. The method preserves a very local property like the DG method, while it has high computational efficiency like the FV method. These advantages facilitate 3D large-scale parallel computations. Numerical computations including a 3D real physical model show its good performance. The method also can be expected to solve other linear hyperbolic equations without essential difficulty.

## Acknowledgments

I appreciate Dr. Y. Zhuang, Prof. Chung, and Dr. L. Zhang very much for their important help and cooperation. This work is supported by the National Natural Science Foundation of China under the grant number 11471328 and 51739007. It is also partially supported by the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.