Numerical Integration for Cylindrical Domain with C2 Function and Point Data

In summary, the conversation discusses the problem of integrating a function w(x,y,z) multiplied by another function V(x,y,z) over a cylindrical domain. The challenge is that V is only known as a set of numerical values at specific points within the domain, and w is a C2 function that is zero on the boundaries and not identically zero inside. The solution proposed is to use Mathematica's interpolation function to approximate V and then numerically integrate the interpolated function. If the data is uniformly spaced, a Riemann sum can also be used. However, since the data is non-uniformly spaced, interpolation is necessary for accurate integration.
  • #1
Dazedandconfu
12
0
ok, I'm not completely sure which section this goes into, but since I'm using this to solve a PDE ima going to put it in here,
I need to integrate w(x,y,z)*V(x,y,z) over a cylindrical domain, it would be fairly simple if V had a "formula" describing the function, but i only have V as a number at a bunch of points(1000 points or so) inside the domain(and on the boundary), w is any C2 function which is zero on the boundaries and not identically zero inside.
I chose a function w and found its values at the same points as V is known, but I'm not sure which is the best way to find an approximation to this integral, any help much appreciated, as always.
 
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  • #2
Mathematica has a function to generate an interpolation of data points. You could use that and then just numerically integrate the interpolated function. See "Interpolation". Otherwise, if your data is uniformly spaced, why not just construct a Riemann sum on the data:

[tex]\iiint\approx \sum_{n=1}^{1000} f(r,\theta,z) \Delta V[/tex]

where:

[tex]\Delta V=r\Delta r\Delta\theta\Delta z[/tex]
 
  • #3
hmm, the data is non uniformly spaced, so the riemann integral cannot be done (without interpolation), so i suppose i'll have to generate a interpolation, and then numerically integrate, thanks!,
 

What is numerical integration?

Numerical integration is a method for approximating the value of a definite integral, which is a mathematical concept that represents the area under a curve. It involves breaking a continuous function into smaller parts and using mathematical algorithms to calculate the area of each part, then summing them up to get an estimated value for the integral.

Why is numerical integration important?

Numerical integration is important because it allows us to solve complex problems that cannot be solved analytically. It is especially useful in scientific and engineering applications where we often encounter functions that are difficult or impossible to integrate using traditional methods.

What are the different types of numerical integration?

There are several types of numerical integration methods, including the trapezoidal rule, Simpson's rule, and the midpoint rule. These methods differ in the way they divide the function and calculate the area of each part. Some methods are more accurate than others, but they all aim to provide a good approximation of the true value of the integral.

How do you choose the best numerical integration method?

The best numerical integration method depends on the specific problem you are trying to solve. Some methods may be more accurate for certain types of functions or for specific intervals of integration. It is important to consider the properties of the function and the desired level of accuracy when choosing a method.

What are the limitations of numerical integration?

Numerical integration has some limitations, such as the accuracy of the approximation depends on the number of intervals used and the complexity of the function. It may also be computationally intensive for highly complex functions. Additionally, numerical integration may not be able to handle certain types of functions, such as those with discontinuities or singularities.

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