Boundary Conditions Initial Conditions

Boundary Conditions & Initial Conditions - by Mandeep Shetty

The words “differential equations (DE)” suggests that it is a type of equation containing derivatives and that is definitely the case, and so we can define ‘differential-equation’ as an equation containing the derivative of one or more dependent variables, w.r.t one or more independent variables. Generally speaking, DEs are classified into two types: If a DE contains only ordinary derivatives of one or more functions with respect to a single independent variable (i.e. the dependent-variable/function is dependent on one independent variable) it is said to be an ordinary-differential-equation (ODE). DE involving only partial derivatives of one or more functions of two or more independent variable (i.e. the dependent-variable/function is dependent on multiple independent variables) is called a partial-differential-equation (PDE). The DEs are extensively used in the mathematical description of a system or a phenomenon (which is called a ‘mathematical model’) as such descriptions usually contain assumptions about the system or phenomenon which frequently involve a rate of change of one or more of the variables involved. Therefore, the mathematical depiction of all these assumptions may be one or more equations involving derivatives.

Since in CFD we deal with a system of PDEs, I am going to concentrate on them in this report. Particularly, the type of PDEs that are extensively used in CFD is second-order linear/quasilinear partial differential equations. We usually come across three-types of second-order PDEs in mechanics. These are classified as elliptic, hyperbolic, and parabolic. When considering a physical system, the way in which changes at one point of a system effect events at other points depends on whether a physical problem represents a steady state or transient phenomenon and whether the propagation speed of disturbances is finite or infinite. This has resulted in a classification of physical behaviors, and hence the classification of PDEs used to model them into elliptic, hyperbolic, and parabolic problems.

Elliptic Equations: These types of equations usually surface when dealing with equilibrium problems and so are the equations used in modeling steady-state situations. The prototypical elliptic equation is Laplace’s equation, which describes an irrotational flow of an incompressible fluid, steady-state heat conduction and also steady, subsonic, inviscid flow etc. An important feature of elliptic problems is that a disturbance in the interior of the solution domain, like a sudden change in temperature at some point in the computational domain, changes the solution everywhere else in the domain and the point itself is influenced by any kind of disturbances/changes at every other point. The disturbance travels throughout the domain at infinite speed and so the disturbance effect all the points in the domain at the same time. Mathematically speaking elliptic equations have no characteristic curves (i.e. they are imaginary) and the solutions are smooth without any discontinuities even when the applied boundary condition may be discontinuous.

Parabolic Equations: These types of equations usually surface when dealing with marching (both time marching and space marching) or propagation problems and so are the equations used in modeling time-dependent problems involving a significant amount of diffusion like transient heat transfer or steady boundary layer flow, unsteady thermal conduction etc. The prototypical parabolic equation is the diffusion equation. An important feature of the parabolic equation is that a disturbance at a point in the interior of the solution domain can only influence events at later in time or region-of-space after that particular point in time or space with the disturbance traveling at infinite speed. This point itself is in turn influenced by events before that point in time or the region-of-space before that point in space (the direction is given by the direction of time marching or space marching). Mathematically speaking parabolic equations have one characteristic curve and the solutions are continuous everywhere but it can support some discontinuity or singularities.

Hyperbolic Equations: These types of equations usually surface when dealing with marching problems and so are the equations used in modeling time-dependent problems like vibrations, steady inviscid supersonic flow, unsteady flows etc. The prototypical hyperbolic equation is the wave equation. An important feature of the hyperbolic equation is that a disturbance at one point in the solution domain can only influence a limited region is space and is the point itself is influenced by changes at certain other regions. This is because the disturbance at any point travels at a finite speed called the wave speed. Mathematically speaking hyperbolic equations have two characteristic line and the solution can contain discontinuities and this is the type of equation we deal with when there are shock-waves in the solution domain.

Generally speaking, different numerical methods are required for a different type of equations. The need for this specialization in numerical approach is rooted in the physics from which the different classes of PDEs arise. By analogy with the conic sections (ellipse, parabola, and hyperbola) partial differential equations have been classified as elliptic, parabolic and hyperbolic. Just as an ellipse is a smooth, rounded object, solutions to elliptic equations tend to be quite smooth. Elliptic equations generally arise from a physical problem that involves a diffusion process that has reached equilibrium, a steady state temperature distribution, for example. The hyperbola is the disconnected conic section. By analogy, hyperbolic equations are able to support solutions with discontinuities, for example, a shock wave. Hyperbolic PDEs usually arise in connection with mechanical oscillators, such as a vibrating string, or in convection driven transport problems. Mathematically, parabolic PDEs serve as a transition from the hyperbolic PDEs to the elliptic PDEs. Physically, parabolic PDEs tend to arise in time-dependent diffusion problems, such as the transient flow of heat in accordance with Fourier\'s law of heat conduction.

[Note: In CFD the flow equations usually don’t fall into any of these categories but we know that the same set of equations, which we call the NS equations, can model different types of flows with different physical behaviors. This is because relative importance of the various terms in NS equations changes in different flow fields and this makes the NS equations ‘act’ as either elliptic, parabolic, hyperbolic equations or a mixture of them (and the solutions have the ‘mathematical behavior’ of one of these types of equation) without actually being different equations]

Having the right PDE which models a system is not enough to get a unique solution for the PDE as differential equations usually have multiple solutions or no solutions. From the perspective of CFD, this is a problem for making appropriate predictions about the flow. For example, if we consider modeling the evolution of a flow using NS equations, multiple solutions mean that the flow can evolve in multiple ways but in reality, the flow progresses in one particular way. So, we need one particular unique solution that tells us how the flow would progress in reality. To get this unique solution we need to specify initial conditions (IC) and boundary conditions (BC) along with the PDEs and the region of space-time on which the PDE required to be satisfied. A boundary condition expresses the behavior of a function on the boundary (border) of its area-of-definition/solution domain. An initial condition is like a boundary condition, but then for the time-direction. Boundary conditions are those which depends on space while initial conditions depend on time. So, in boundary condition space coordinate will be varied and in initial conditions, time will be varied. A boundary value problem (BVP) has conditions specified at the extremes (\"boundaries\") of the independent variable in the equation whereas an initial value problem (IVP) has all of the conditions specified at the same value of the independent variable. To be more specific, differential equations can be either boundary value problems or initial value problems. BVPs require only BCs because they are generally elliptical in nature (propagation of disturbances everywhere almost instantaneously) while IVPs require both ICs & BCs because they can be either parabolic or hyperbolic in nature (propagation along characteristics).

There are four broad classes of boundary conditions:

• Dirichlet boundary conditions: The value of the dependent variable is specified on the boundary.
• Neumann boundary conditions: The normal derivative of the dependent variable is specified on the boundary.
• Cauchy boundary conditions: Both the value and the normal derivative of the dependent variable are specified on the boundary.
• Robin boundary conditions: The value of a linear combination of the dependent variable and the normal derivative of the dependent variable is specified on the boundary.

Each class of PDE\'s requires a different class of boundary conditions in order to have a unique, stable solution.

1) Elliptic equations require either Dirichlet or Neumann boundary conditions on a closed boundary surrounding the region of interest. Other boundary conditions are either insufficient to determine a unique solution, overly restrictive, or lead to instabilities.

2) Hyperbolic equations require Cauchy boundary conditions on an open surface. Other boundary conditions are either too restrictive for a solution to exist, or insufficient to determine a unique solution.

3) Parabolic equations require Dirichlet or Neumann boundary conditions on an open surface. Other boundary conditions are too restrictive.