Numerical methods for partial differential equations facts for kids

Numerical methods for partial differential equations are like special computer tools. They help us find approximate answers to very complex math problems called partial differential equations (PDEs). These equations describe how things change over time and space, like how heat spreads or how water flows. Since it's often impossible to solve these equations perfectly with just math, we use computers to get really good estimates.

Software Tools for Solving PDEs

Many computer programs and libraries help scientists and engineers solve PDEs.

• Chebfun is a well-known software used in this area.
• Other tools are based on a method called the finite element method, like:
  • FreeFem++
  • FEniCS Project

Why We Need Numerical Methods

Solving Tough Science Problems

Many important PDEs come from studying physics and other sciences. Scientists and mathematicians have tried for a long time to solve them perfectly using only math. But for most PDEs, there isn't a simple mathematical way to find an exact answer. This is why we need numerical methods. They give us powerful ways to find very close answers using computers.

The Finite Difference Method (FDM)

One of the oldest and simplest ways to solve PDEs with computers is the finite difference method (FDM). This method works by turning the continuous changes in an equation into small, separate steps. Imagine you want to find the speed of a car. Instead of knowing its exact speed at every moment, FDM would look at its position at two very close times and calculate the average speed between those points.

FDM is good for simpler problems. But it struggles with more complicated equations, especially those that are "non-linear" (meaning they don't follow a simple straight line pattern), like the Navier–Stokes equations which describe fluid movement. Because of these difficulties, experts started looking for new methods, such as the finite element method (FEM). Others worked on making FDM better.

Improving the Finite Difference Method

Scientists have found ways to improve FDM so that it keeps important properties of the original PDE.

Methods That Preserve "Integrability"

Some mathematicians, like Ryogo Hirota, developed methods that keep a special mathematical property called integrability. This property is important in understanding how systems change over time. These improved methods are often more accurate than the original FDM.

Methods That Preserve Physical Properties

Many PDEs come from physics, describing things like energy or momentum. So, scientists developed "difference methods" that keep these physical properties intact. These are called structure preserving numerical methods. Here are some examples:

• Symplectic integrators
• Discrete gradient method
• Discrete variational derivative method (DVDM)

Some experts are also looking at how these methods connect with numerical linear algebra, which is about solving systems of equations.

Other New FDM Approaches

While the methods above are very accurate, they can only be used for certain types of PDEs. Because of this, new kinds of FDM are still being developed. Some examples include:

• Shortley-Weller approximation
• Swarztrauber-Sweet approximation
• Ascher-Mattheij-Russell difference formula

Checking Computer Solutions for PDEs

Besides finding approximate solutions, scientists also work on proving that a solution found by a computer is actually real and not a "phantom solution" (a fake one). This is important because sometimes computers can give answers that look right but don't actually exist in the real mathematical problem. Cases where this has happened have been reported.

PDEs Studied for Validated Numerics

Here are some of the PDEs that have been checked using these "validated numerics" methods:

Experts in This Field

Many brilliant minds have contributed to the field of numerical methods for PDEs. Some of them include:

• Israel Gelfand
• John von Neumann
• Masatake Mori
• Peter Deuflhard
• Peter Lax
• Shinichi Oishi