# Mode-dependent finite-difference discretization of linear homogeneous differential equations

## Abstract

A new methodology utilizing the spectral analysis of local differential operators is proposed to design and analyze mode-dependent finite-difference schemes for linear homogeneous ordinary and partial differential equations. The authors interpret the finite-difference method as a procedure for approximating exactly a local differential operator over a finite-dimensional space of test functions called the coincident space, and show that the coincident space is basically determined by the nullspace of the local differential operator. Since local operators are linear and approximately with constant coefficients, the authors introduce a transform domain approach to perform the spectral analysis. For the case of boundary-value ordinary differential equations, (ODEs), a mode-dependent finite-difference scheme can be systematically obtained. For boundary-value partial differential equations (PDEs), mode-dependent 5-point, rotated 5-point, and 9-point stencil discretizations for the Laplace, Helmholtz, and convection-diffusion equations are developed. The effectiveness of the resulting schemes is shown analytically, as well as by considering several numerical examples.

- Authors:

- Publication Date:

- Research Org.:
- Dept. of Mathematics, Univ. of California, Los Angeles, CA (US); Dept. of Electrical Engineering and Computer Science, Univ. of California, Davis, CA (US)

- OSTI Identifier:
- 6450271

- Resource Type:
- Journal Article

- Journal Name:
- SIAM J. Sci. Stat. Comput.; (United States)

- Additional Journal Information:
- Journal Volume: 9:6

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; FINITE DIFFERENCE METHOD; LAPLACE EQUATION; PARTIAL DIFFERENTIAL EQUATIONS; BOUNDARY-VALUE PROBLEMS; CONVECTION; DIFFERENTIAL EQUATIONS; DIFFUSION; HELMHOLTZ THEOREM; NUMERICAL SOLUTION; ENERGY TRANSFER; EQUATIONS; HEAT TRANSFER; ITERATIVE METHODS; MASS TRANSFER; 990200* - Mathematics & Computers

### Citation Formats

```
Kuo, C C, and Levy, B C.
```*Mode-dependent finite-difference discretization of linear homogeneous differential equations*. United States: N. p., 1988.
Web. doi:10.1137/0909069.

```
Kuo, C C, & Levy, B C.
```*Mode-dependent finite-difference discretization of linear homogeneous differential equations*. United States. https://doi.org/10.1137/0909069

```
Kuo, C C, and Levy, B C. Tue .
"Mode-dependent finite-difference discretization of linear homogeneous differential equations". United States. https://doi.org/10.1137/0909069.
```

```
@article{osti_6450271,
```

title = {Mode-dependent finite-difference discretization of linear homogeneous differential equations},

author = {Kuo, C C and Levy, B C},

abstractNote = {A new methodology utilizing the spectral analysis of local differential operators is proposed to design and analyze mode-dependent finite-difference schemes for linear homogeneous ordinary and partial differential equations. The authors interpret the finite-difference method as a procedure for approximating exactly a local differential operator over a finite-dimensional space of test functions called the coincident space, and show that the coincident space is basically determined by the nullspace of the local differential operator. Since local operators are linear and approximately with constant coefficients, the authors introduce a transform domain approach to perform the spectral analysis. For the case of boundary-value ordinary differential equations, (ODEs), a mode-dependent finite-difference scheme can be systematically obtained. For boundary-value partial differential equations (PDEs), mode-dependent 5-point, rotated 5-point, and 9-point stencil discretizations for the Laplace, Helmholtz, and convection-diffusion equations are developed. The effectiveness of the resulting schemes is shown analytically, as well as by considering several numerical examples.},

doi = {10.1137/0909069},

url = {https://www.osti.gov/biblio/6450271},
journal = {SIAM J. Sci. Stat. Comput.; (United States)},

number = ,

volume = 9:6,

place = {United States},

year = {1988},

month = {11}

}