universal data

Chapter 241
"Preprocessing method for joint compact difference discrete equations of two-dimensional Helmholtz equation"!
This is the full name of this project.Very long and mouth-wrenching.

After hearing this, Cheng Nuo took a deep breath, "This project is very difficult!"

The so-called Helmholtz equation, also known as the Helmholtz equation.

It is an elliptic partial differential equation that describes electromagnetic waves. It is named after the German physicist Helmholtz and usually appears in the study of physical problems involving partial differential equations that have both space and time dependence.

Because of its relationship to the wave equation, the equation is widely used in physics in related research fields such as electromagnetic radiation, seismology, and acoustics.

The equation can be expressed as △f(x)+k^2f(x)=g(x), x∈Ω.

Among them, △ is the Laplacian operator, k is the wave number, the closed set Ω∈R^d, d=1, 2, 3...

There are many numerical solutions to the Helmholtz equation, mainly including finite difference method, finite element method, boundary element method and meshless method.However, the above requirements for derivatives are relatively high and the amount of calculation is relatively large.

Therefore, this research project is to discretize the Helmholtz equation through the joint compact difference scheme (CCD).Combining the values ​​of the Helmholtz polynomial at each point and the values ​​of the two adjacent points and the first and second order derivative values, and then combining the Taylor expansion to derive a linear system.

To put it simply, this is a very complex and extremely computationally demanding project.

Compared with some practice questions Cheng Nuo and the others saw, they were not at the same level of difficulty at all.That's why it is necessary to declare a topic specifically and spend a lot of time researching this issue.

In response to Cheng Nuo's exclamation, Liao Zhixing nodded with a smile, "Indeed, this project is indeed somewhat difficult for you at present. However, it is not impossible to complete."

"What's more, it doesn't require you to complete this task alone. It depends on the cooperation of the small team of four of you. Each of you should play to your own strengths. Study hard."

Finally, Liao Zhixing said in a serious voice, "You must always keep in mind one sentence: If there is a will, there will be a way. If the boat is broken, the two hundred Qin Pass will eventually belong to Chu. Those who work hard will live up to the sky. If you try your best, you can swallow Wu with three thousand! "

After pouring a bowl of chicken soup for the four of them, Liao Zhixing picked up the thermos cup in front of the table and took a sip.

He raised his wrist and looked at the time on his watch.Then get up and stand up.

"Okay, I still have something to do here. I will leave this to you. Jiang Shuobo, Wang Genji, and Zhuang Poxiao, please introduce the main content of this project and your current research progress to Cheng Nuo. Then tell Cheng Nuo Nuo find something to do. You can discuss it yourself as to what to do. "

"Okay." The three seniors nodded in response, watching Liao Zhixing push the door and leave.

In the office, the atmosphere was silent for a few minutes.

Jiang Shuobo, a senior senior, smiled to ease the atmosphere, "Everyone, don't be so silent. Genji, please tell this junior Cheng Nuo about the project we are working on. Let him have a general understanding of what we are doing."

"No problem." Wang Genji snapped his fingers, took out his laptop from his schoolbag, connected it to the projector in the office, raised his eyebrows at Cheng Nuo, "Junior Cheng Nuo, look at your confused little eyes, It must have been a dark eye for this subject."

"But it doesn't matter, you are only a freshman, and your knowledge reserve is obviously not enough. You don't need to understand him thoroughly on this subject. For the difficult ones, leave it to us three. You just need to help us."

"I have a senior here, and I promise to take you to pretend to fly!" Wang Genji patted his chest, and said this sentence full of pretense.

"Genji, stop talking nonsense, hurry up and talk. Then we will leave after assigning tasks." Jiang Shuobo adjusted his glasses and urged.

"Okay, now, now!" Wang Genji manipulated the mouse and clicked to open a Word document.

He coughed lightly, and said slowly, "Our project, in fact, is simply to use the compact difference format to discretize the two-dimensional Helmholtz equation. The difference format has sixth-order precision, three-point difference and implicit specialty……"

Halfway through the sermon, Wang Genji seemed to suddenly remember something, and asked Cheng Nuo, "Junior Cheng Nuo, you should know what the Helmholtz equation and the compact difference scheme are, right? These two terms seem to be appropriate for you. Haven't gotten in touch yet."

"However, since Mr. Liao asked you to join our research group, it shouldn't be a problem for you, right?" Wang Genji looked at Cheng Nuo with a smile.

"Helmholtz equation and compact difference scheme?" Cheng Nuo scratched his head and said modestly, "I've heard of it."

"Oh?" Wang Genji's eyes lit up, and he stretched out his hand to signal Cheng Nuo, "Then Junior Brother Cheng Nuo, please briefly explain your understanding. Just taking this opportunity, the senior will add to you what you have missed. Know these two nouns."

In Wang Genji's opinion, with Cheng Nuo's freshman education, it is not easy to know these two obviously super-class nouns.Probably just heard it by accident.In the end, it's not up to you to explain yourself.

Just in time, Wang Genji planned to take this opportunity to establish the prestige of the senior in Cheng Nuo's heart.

"Then I would like to thank you senior. I will trouble you." Cheng Nuo said sincerely.

"Hey, no trouble, no trouble." After being praised by Cheng Nuo, Wang Genji waved his hand and laughed triumphantly, "It's our job as seniors to guide students in their studies. Just say it boldly, It doesn't matter if you're wrong!"

Cheng Nuo nodded, "Then let me talk about the Helmholtz equation first. It was named by the German physicist Helmholtz. It refers to the hyperbolic partial equation in mathematics with the form of (△2+k2)ψ=f Differential equation,...where △2 is the Laplace operator, which is the Helmholtz equation in the Cartesian coordinate system; ψ is the function to be found; k2 is a constant; f is the source function. When When f is equal to zero, it is called a homogeneous Helmholtz equation; when f is not equal to zero, it is called an inhomogeneous Helmholtz equation..."

Cheng Nuo talked eloquently: "...When a function F(x, y, z, t) undergoes simple harmonic changes with time, it can be expressed in the form of F(x, y, z)ejwt, then д/дt Equivalent to jω, д2/дt2 is equivalent to -ω2, substitute..."

From the source, formula, solution, transformation, and application of the Helmholtz equation, Cheng Nuo can be said to cover everything, detailing everything in every detail, and can't find any mistakes.

Just the term Helmholtz equation, Cheng Nuo talked on and on for more than ten minutes. What he said was full of spit and cadence, and it still hasn't stopped.

Wang Genji, who was sitting opposite Cheng Nuo, had already opened his mouth unbelievably.

Have heard of it?

I just believed your nonsense because I believed in your evil at the beginning.

This is called a little heard, so what are we?
This guy, is there any misunderstanding about this word?
(End of this chapter)