Partial Differential Equations

Partial differential equations (PDE's) are one of the most fundamental applications of mathematics. They describe phenomena of the physical, natural and social science such as fluids, gravitational and electromagnetic fields, and the human body. They also play an important role in fields like aircraft simulation, computer graphics, and weather prediction.

There exist many partial differential equations, but from the view point of mathematics, three important examples are Laplace's equation, diffusion equation, and wave equation. The general linear 2 dimensional equation of the PDE's is can be written as follows

$$a\frac{\partial^2V}{\partial x^2} + 2b\frac{\partial^2V}{\partial... ... + d\frac{\partial V}{\partial x} + e\frac{\partial V}{\partial y} + fV + g = 0$$ (1)

where it is Laplace equation if $b^2 < ac$, Wave equation if $b^2 > ac$, and Diffusion/Schrödinger equation if $b^2 = ac$.

In this paper we will perform benchmarks on the Laplace's equation and diffusion equation.

Elliptic equation like Laplace equation is commonly found in multidimensional steady state problems. We can write the simplest form of elliptic equation (Poisson equation) by simplifying the equation (6.1) as follows

$$\frac{\partial^2V}{\partial x^2} + \frac{\partial^2V}{\partial y^2} = \rho_{x,y}$$ (2)

This equation can also be rewritten in finite-difference form using the function $u(x,y)$ and its representation values at the discrete set of points

$$x_{j} = x_{0} + j\Delta \;\;\; and \;\;\; y_{l} = y_{0} + l\Delta \;\;\;$$ (3)

with the indices $j=0,1,2,...,J$ and $l=0,1,2,...,L$. Finite-difference representation (equation (6.4)) of equation 6.2 and its equivalent format (equation (6.5)) which can be written as a system of linear equations in matrix form can be described as follows

$$\frac{u_{j+1,l} - 2u_{j,l} + u_{j-1,l}}{\Delta^2} + \frac{u_{j,l+1} - 2u_{j,l} + u_{j,l-1}}{\Delta^2} = \rho_{j,l}$$ (4)

$$u_{j+1,l} + u_{j-1,l} + u_{j,l+1} + u_{j,l-1} - 4u_{j,l} = \Delta^2\rho_{j,l}$$ (5)