Numerical Web

Category: Numerial Theories

  • Open Source Modelling and Visualization Libraries

    Introduction Open source 3D modeling and visualization refers to the use of software and tools that are freely available and can be modified and distributed by anyone. These tools and libraries provide a wide range of functionality for 3D modeling and visualization, including the ability to create and edit 3D models, perform numerical simulations, and…

  • NGSOLVE Tutorial (4) Static Condensation Part B. Inhomogeneous Dirichlet B.C and Automatic utility

    Inhomogeneous Dirichlet Boundary Conditions In case of inhomogeneous Dirichlet boundary conditions, we combine the technique of Dirichlet data extension with the above static condensation principle in the following code. The code you provided is solving a Poisson equation with a manufactured solution and then calculates the error between the manufactured solution and the numerical solution…

  • NGSOLVE Tutorial (4) Static Condensation Part A

    Descriptions Static condensation is a technique used in the solution of partial differential equations (PDEs) to reduce the size of the system to be solved. In the context of the ngsolve example, it likely refers to the use of static condensation to reduce the size of the system of equations being solved by the solver…

  • Proper Generalized Decomposition(PGD)

    Proper Generalized Decomposition (PGD) is a mathematical technique that can be used to approximate the solution of partial differential equations (PDEs) by expanding it in a basis of simple functions. It’s a generalization of the Proper Orthogonal Decomposition (POD) method, which can be used when the solution of the PDE is not linear. PGD is…

  • Proper Orthogonal Decomposition(POD)

    Proper Orthogonal Decomposition (POD) is a mathematical technique that can be used to extract the most important modes of a system from a set of snapshots of the system’s state. It is a data-driven method and it can be used to identify the most important features of a system regardless of whether the system is…

  • Finite Difference Scheme (4) Python Examples for ODE

    Here is an example of how to use NumPy to solve a second-order ODE using finite differences: First, we can import NumPy and set up the grid of x-values: Next, we can define the function $y(x)$ and its second derivative $y”(x)$: Note that in this example, we are using a function that has a known…

  • Finite Difference Scheme (3) Examples for Ordinary Differential Equation

    Here is an example of how to solve a second-order ordinary differential equation (ODE) using finite differences: Suppose we have the following second-order ODE: where $y$ is a function of $x$ and $y”$ is the second derivative of $y$ with respect to $x$. To solve this ODE using finite differences, we can start by discretizing…

  • Finite Difference Scheme (2) Second order schemes

    Here are examples of numerical schemes for finite differences that have second-order accuracy: Forward differences: To achieve second-order accuracy using forward differences, we can use the following formula: This formula uses three function values to approximate the derivative, which results in a higher-order approximation. For example, suppose we have the function $f(x) = x^2$ and…

  • Finite Difference Scheme (1) Basics

    A numerical scheme for finite differences is a mathematical way to approximate a derivative of a function using finite differences, which are small values that represent the difference between two nearby points on a grid. This allows us to approximate the derivative of a function by using a finite set of data points rather than…