"Preprocessing Method for Joint Compact Difference Separation Equations of Two-Dimensional Helmholtz Equations"!

That's the full name of this project. It's very long and tongue-twisting.

After hearing this, Cheng Nuo gasped, "This project is very difficult!"

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

is an elliptic partial differential equation that describes electromagnetic waves, named after the German physicist Helmholtz, and is commonly found in the study of physics problems involving partial differential equations that are dependent on both space and time.

Because of its relationship to the wave equation, it has a wide range of applications in physics, such as electromagnetic radiation, seismology, and acoustics.

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

where △ is the Laplace operator, k is the wave number, and the closed set Ω∈R^d, d=1,2,3......

There are many numerical solutions to Helmholtz's equations, including finite difference method, finite element method, boundary element method and meshless method. However, the above ones have high requirements for derivatives and are relatively computationally intensive.

Therefore, this research project aims to discretize the Helmholtz equation by combining the compact difference format (CCD). Combine the values of the Helmholtz polynomial at two extremely adjacent points at each point with the values of the first and second derivatives to derive the linear system in combination with the Taylor expansion.

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

Compared with some of the practice questions that 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 and spend a lot of time researching this problem.

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

"What's more, you don't have to do this alone. It's up to the four of you to work together as a small team. Each plays to their strengths. Immerse yourself in research. ”

In the end, Liao Zhixing said in a serious voice, "You must always keep in mind one sentence: where there is a will, there is a way, the cauldron is broken, and the two hundred Qin Passes will eventually belong to Chu." Hard-working people, the sky is not worth it, lying on the salary and tasting the gall, three thousand Yuejia can swallow Wu!"

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'll leave it to you here. Jiang Shuobo, Wang Genji, and Zhuang Poxiao, please introduce to Cheng Nuo the main content of this topic and your current research progress. Then find something to do for Cheng Nuo. What to do, you can discuss it yourself. ”

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

Inside the office, the atmosphere was silent for several minutes.

Senior Jiang Shuobo smiled softly, "Don't be so silent. Foundation, you tell this Cheng Nuo junior about the topic we are working on. Give him an idea of what we're doing. ”

"No problem. Wang Genji snapped his fingers, took out his laptop from his schoolbag, connected it to the projector in the office, and raised his eyebrows at Cheng Nuo, "Senior Brother Cheng Nuo, looking at your confused little eyes, you must be blackened by this subject." ”

"But it doesn't matter, you're only a freshman, and your knowledge is obviously not enough. You don't need to be able to understand him thoroughly. The difficult ones, leave it to the three of us. You just have to give us a hand. ”

"I'm here with a senior, I promise to take you to pretend to be forced to fly with you!" Wang Genji patted his chest and said this sentence with a pretending to be full of force.

"Foundation, don't talk nonsense, you hurry up. Then we assign the task and disperse. Jiang Shuobo held his glasses and urged.

"Okay, right away, right away!" Wang Genji manipulated the mouse and clicked on an ORD document.

He coughed lightly and spoke slowly, "Our project, in simple terms, is to discretize the two-dimensional Helmholtz equation in the format of compact difference. The differential format has the characteristics of sixth-order precision, three-point difference, and implicit......"

Halfway through the sermon, Wang Genji seemed to suddenly remember something, and asked Cheng Nuo, "Senior Brother Cheng Nuo, you should know what the Helmholtz equation and the compact difference format are, right? ”

"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 equations and compact difference formats?" Cheng Nuo scratched his head and said modestly, "I've heard a little." ”

"Oh?" Wang Genji's eyes lit up, and he stretched out his hand to signal Cheng Nuo, "Then Senior Brother Cheng Nuo, you can briefly explain your understanding." Just take this opportunity to add to you what you missed, so that you know these two terms thoroughly. ”

In Wang Genji's view, with Cheng Nuo's freshman education, it is not easy to know these two obviously out-of-class terms. I guess I just heard about it by chance. In the end, it's not up to you to explain.

It just so happened that Wang Genji planned to take this opportunity to establish the prestige of his senior in Cheng Nuo's heart.

"Then thank you so much, senior. Excuse me. Cheng Nuo's tone was sincere.

"Hey, no trouble, no trouble. After receiving Cheng Nuo's praise, Wang Genji waved his hand and laughed proudly, "Guiding juniors to study is our duty as seniors." You can say it boldly, even if it's wrong!"

Cheng Nuo nodded, "Then let me talk about the Helmholtz equation first, which was named by the German physicist Helmholtz." It refers to a hyperbolic partial differential equation that has the form of (△2+k2)ψ=f mathematically,...... where △2 is the Laplace operator, in the Cartesian coordinate system, is the Helmholtz equation, and ψ is the function to be solved. k2 is a constant, and f is the source function. When f is equal to zero, it is called the homogeneous Helmholtz equation; When f is not equal to zero, it is called the non-homogeneous Helmholtz equation ......"

Cheng Nuo said eloquently: "...... When a function F(x,y,z,t) changes in simple harmony with time, it can be expressed in the form of F(x,y,z)ejt, where д/дt is equivalent to jω, and д2/дt2 is equivalent to -ω2, substituting ......"

From the source, formula, solution, transformation, and application of the Helmholtz equation, Cheng Nuo can be said to be comprehensive, and every place is meticulous, and he can't find any mistakes.

Just the term Helmholtz equation, Cheng Nuo talked endlessly for more than ten minutes, and what he said was called a spittle, subdued and frustrated, and there was no tendency to stop until now.

Wang Jiji, who was sitting opposite Cheng Nuo, had grown his mouth in disbelief at this time.

Heard of it?

I believed your evil in the first place, so I believed your nonsense.

If we have heard of it, then what are we?

This guy, is there any misunderstanding of this word?