PHY 604: Computational Methods in Physics and Astrophysics II

PDEs

PHY 604: Computational Methods in Physics and Astrophysics II

On to PDEs...

● Next up: PDEs

– PDEs are at the heart of many physical systems

– We will study three classes of PDEs, represented by the wave/advection equation, the Poisson equation, and the diffusion equation

● Where do we stand?

● Differentiation:

– We saw how Taylor expansions give rise to difference formula with varying orders of accuracy

– These ideas will be at the heart of the spatial discretization we use with PDEs

● Interpolation:

– We will see the interpolation ideas again as we reconstruct our discretized data to find values at interfaces in our PDE discretizations

PHY 604: Computational Methods in Physics and Astrophysics II

On to PDEs...

● ODEs:

– A common procedure is to spatially discretize a PDE and then solve the result initial value ODE system using ODE methods—this is called the method of lines approach

● Linear algebra:

– We will have a choice of discretizing explicitly or implicitly. Implicit discretizations often result in a linear system to solve, using our linear algebra techniques

– We will see the iterative methods come into play when we consider the Poisson equation

● FFTs:

– As already motivated, FFTs can be used to transform a PDE into an algebraic equation in Fourier-space, enabling its easy solution. (Some restrictions apply)

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

● Different types of PDEs require different solution methods

– Methodology matches the underlying behavior

● We will study the three main classes of PDEs:

– Hyperbolic

● e.g. the wave equation: ut - uxx = 0

● Characterized by real, distinct propagation speeds (for a system, the eigenvalues are real)

● Requires initial conditions and boundary conditions (on the upwind boundaries)

● Supports discontinuities, propagation of waves, etc.● The equations of compressible hydrodynamics form a hyperbolic system● The scalar linear advection equation will be our model for these methods

PHY 604: Computational Methods in Physics and Astrophysics II

The Sod problem from compressible hydrodynamics—here we see three characteristic waves that carry information propagating outward from an initial disturbance

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

– Elliptic

● e.g. the Poisson equation: uxx + uyy = f

● No “speed” associated with propagating information—the solution responds instantaneously to the boundaries

● Requires boundary conditions on all sides● Solutions are smooth● Also arises in low-Mach number flow as a constraint on the velocities,

electrostatics, gravity, ...

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

with u = 0 on the boundary.

This example comes from a simple python code on my website

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

– Parabolic

● e.g. the diffusion equation: ut = uxx

● Inbetween hyperbolic and elliptic● Solution smooths out sharp features

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

Diffusion of a Gaussian in one-dimension

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

● Formally, these names come from a mathematical classification analogous to conic sections

– Written as:

– Hyperbolic if

– Parabolic if

– Elliptic if

(Wikiped

ia)

PHY 604: Computational Methods in Physics and Astrophysics II

Introduction to PDEs

● We can also discuss them in terms of initial value vs. boundary value problems

– Typical boundary conditions are

● Dirichlet (specify value at the boundary)● Neumann (specify the derivative normal to the boundary)● Periodic

● Many equations or systems are mixed types, e.g.

– Viscous flow (both hyperbolic + parabolic)

– Constrained flow (hyperbolic + elliptic)

PHY 604: Computational Methods in Physics and Astrophysics II

Discretization

● We need to discretize both space and time (or multiple space dimensions)

● There are many different types of spatial discretizations we can employ

– Finite difference

– Finite volume

– Finite element

– Spectral

– Particle methods (SPH)

● Each have their own strengths and weaknesses

● All but finite-element see wide use in astrophysics

PHY 604: Computational Methods in Physics and Astrophysics II

Spectral Methods

● Solution is expressed in terms of a superposition of globally-defined basis functions (Fourier series, Chebyshev, or Legendre functions)

● Solve for evolution of the weights in the superposition

● Ideally for smooth flow

– Doesn't handle discontinuities (e.g. shocks) well

● Wide application in turbulence studies

PHY 604: Computational Methods in Physics and Astrophysics II

Finite Elements

● Derivation similar to spectral—form inner product of function and basis functions

– In F-E, compact basis functions are used

● Popular in engineering—works well on irregular domains

Wikipedia

PHY 604: Computational Methods in Physics and Astrophysics II

Structured Grids

● Discretization on a series of logically Cartesian grids

– Allows for easy access to any element, avoiding indirect addressing on machines—maximizes cache performance

● Two popular approaches: finite-difference and finite-volume

– Underlying mathematical formulation differs

– Often can result in identical discretizations

PHY 604: Computational Methods in Physics and Astrophysics II

Finite Difference Approximation

● Function values are stored at discrete grid points

– Discrete values are simply:

– Replace derivatives in the PDE with discrete differences between neighboring grid points

– This is basically what we've already done with ODEs (think about the orbit problem)

PHY 604: Computational Methods in Physics and Astrophysics II

Finite Volume Approximation

● In the finite-volume approach, we store the average of the function value over an interval/zone

– Where:

– Note that half-integers are used to label the zone boundaries

– This discretization arises naturally when we model conservation laws in integral form

PHY 604: Computational Methods in Physics and Astrophysics II

Multiple Dimensions

● Either of these discretizations can be extended to multiple dimensions, using similar definitions

2-d finite-difference grid 2-d finite-volume grid

PHY 604: Computational Methods in Physics and Astrophysics II

Structured Adaptive Mesh Refinment

● Structured AMR uses a nested hierarchy of grids to focus resolution in regions of complex flow

PHY 604: Computational Methods in Physics and Astrophysics II

SPH

● No grid

● Represent mass distribution as a collection of particles

● Form continuous quantities by integrating over particles via a smoothing kernel

● Lagrangrian approach (more on that later...)

Two spherical mass distributions represented by a structured grid (a) and SPH particles (b). Note, there is a lot of unneeded resolution in the space between the spheres in the grid case.

(Stephen Oxley)

PHY 604: Computational Methods in Physics and Astrophysics II

Aside: Gravity Only

● Pure gravity simulations often employ an N-body technique

– Particles represent mass

– Accelerations are computed by computing the total gravitational force on each particle due to all other particles

– Position/velocity advanced via Newton's laws

● Not a PDE

● Direct calculation is N2, but we'll see later there is an N log N method

PHY 604: Computational Methods in Physics and Astrophysics II

Grid Basics

● It should not come as a surprise that the finer we make the grid, the more accurate our solution

● For time-dependent problems (like hyperbolic or parabolic equations), sometimes there is a relation between the size of the timestep we can take and the grid spacing

● Now when we talk about order-of-accuracy, we will need to discuss both the spatial and temporal accuracy