Partial differential equations (PDEs)

10. Partial differential equations (PDEs)#

Partial differential equations occur when we are dealing with functions of more than one variable, for instance fields.

Examples:

  • Electrostatic potential \(\phi (x,y,z)\) (Poisson’s equation)

\[ \Delta\phi(x,y,z) = -\frac{\rho(x,y,z)}{\epsilon_0} \]

  • Density or temperature profiles (diffusion/heat equation)

\[ \frac{\partial u(\mathbf{x},t)}{\partial t} = D \Delta u(\mathbf{x},t) \]

  • Displacement (amplitude) profile (wave equation)

\[ \frac{\partial^2 u(\mathbf{x},t)}{\partial t^2} = c^2\Delta u(\mathbf{x},t) \]

  • Fluid dynamical fields (flow velocity) – e.g. Navier-Stokes equations

General methods for solving PDE’s#

  • Finite difference method

    • Approximate the derivatives by finite differences

    • Easier to implement than other methods

    • Works best for regular (rectangular) shapes

  • Finite element method

    • Subdivide the system into smaller parts – finite elements

    • Boundary value problems in 2/3 dimensions

    • Works well for irregular shapes

  • Finite volume method

    • Convert surface integrals around each mesh point into volume integrals

    • Conserves mass by design, good for fluid dynamical equations

Here we will focus on the finite difference method.

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