Classification of Partial Difference Equations

Classification of Partial Differential Equations (PDEs)

1. Introduction to PDEs and Their Classification

Partial Differential Equations (PDEs) are mathematical tools that govern many important physical processes in nature [1]. Understanding their characteristics is crucial in computational fluid dynamics (CFD). The behaviour of PDEs is classified based on their mathematical properties, which, in turn, have significant physical implications .

2. Physical Classification of Problems Governed by PDEs

PDEs can be broadly categorised based on the type of physical problems they describe :

  • Equilibrium Problems (Boundary Value Problems):
    • In these problems, a solution of a given PDE is sought within a closed domain, subject to a prescribed set of boundary conditions .
    • The solution at every point in the domain depends upon the specified boundary condition at *every point* on its boundary, hence they are sometimes called "jury problems" .
    • Mathematically, equilibrium problems are governed by elliptic PDEs .
    • Examples: Steady-state temperature distributions (Laplace's equation), incompressible inviscid flows, and equilibrium stress distributions in solids.
  • Marching Problems (Initial Value or Initial Boundary Value Problems):
    • For these problems, the solution of a PDE is required on an open domain, subject to a set of initial conditions and a set of boundary conditions.
    • The solution must be computed by marching outward from the initial data surface while satisfying the boundary conditions .
    • Mathematically, marching problems are governed by either hyperbolic or parabolic PDEs .
    • Examples: Unsteady inviscid flow, steady supersonic inviscid flow, transient heat conduction, and boundary-layer flow .

3. Mathematical Classification of PDEs

The mathematical classification of PDEs is based on the concept of characteristics, which are lines (in 2D) or surfaces (in 3D) where certain properties remain constant or derivatives may be discontinuous . These characteristics are related to the directions in which "information" can be transmitted in a physical problem .

For a general second-order PDE with two independent variables, it can be written as :

a∂2φ/∂x2 + b∂2φ/∂x∂y + c∂2φ/∂y2 = H(φ, ∂φ/∂x, ∂φ/∂y, x, y)

The classification depends only on the coefficients of the second-derivative terms ($a$, $b$, and $c$) . The discriminant ($b^2 - 4ac$) determines the type of the PDE :

  • Hyperbolic: If $b^2 - 4ac > 0$ .
  • Parabolic: If $b^2 - 4ac = 0$ .
  • Elliptic: If $b^2 - 4ac < 0$ .

Note that if $a$, $b$, and $c$ are not constants, the classification may change from point to point within the problem domain . Any real nonsingular transformation does not change the type of PDE.

4. Types of PDEs

4.1. Hyperbolic PDEs

  • Definition: A PDE is hyperbolic if the discriminant $b^2 - 4ac$ is positive .
  • Characteristics: Two distinct families of real characteristic curves exist. Along these curves, second derivatives are not uniquely determined, and discontinuities in the highest-order derivatives (e.g., shock waves) may appear .
  • Physical Behaviour: Hyperbolic equations admit wave-like solutions . Information propagates along these characteristics, leading to limited zones of influence where the solution at a given point depends only on data within a specific region (domain of dependence) . They are associated with "marching problems" .
  • Canonical Form: Can be reduced to forms like φξη = h2(...) or φξξ - φηη = h1(...) by a coordinate transformation using the characteristic curves .
  • Examples:
    • Wave equation: utt = c2uxx . The first-order linear convection equation, ut + c ux = 0, is also used as a model hyperbolic equation .
    • Unsteady inviscid flow and steady supersonic inviscid flow . An example includes steady supersonic inviscid flow over a blunt body where the flow region near the nose is elliptic and farther downstream it is hyperbolic .
  • Well-Posed Problems: For hyperbolic equations, initial conditions are required on the unknown function and its first derivatives along a curve . The curve must not coincide with a characteristic; otherwise, a unique solution cannot be obtained, and the problem is "ill-posed" . The "domain of dependence" and "region of influence" are crucial concepts for hyperbolic problems .
  • Numerical Stability: The Courant-Friedrichs-Lewy (CFL) condition is a numerical stability requirement for many explicit methods used to solve hyperbolic PDEs, stating that the analytic domain of influence must lie within the numerical domain of influence .

4.2. Parabolic PDEs

  • Definition: A PDE is parabolic if the discriminant $b^2 - 4ac$ is zero .
  • Characteristics: Only a single family of characteristic curves exists 
  • Physical Behaviour: Parabolic PDEs are associated with diffusion processes . Their solutions correspond to damped waves . Unlike hyperbolic equations, the solution of a parabolic equation at a given time depends upon the entire physical domain (i.e., all initial and side boundary conditions up to that time/location) . They are associated with "marching problems".
  • Canonical Form: Can be written as φξξ = g(...) .
  • Examples:
    • Heat equation (or diffusion equation): ∂U/∂t = α ∂2U/∂x2 . This governs transient heat conduction .
    • Boundary-layer flow . The "parabolized" Navier-Stokes equations are also discussed .

4.3. Elliptic PDEs

  • Definition: A PDE is elliptic if the discriminant $b^2 - 4ac$ is negative.
  • Characteristics: No real characteristics exist for elliptic equations .
  • Physical Behaviour: Elliptic PDEs govern equilibrium problems or "jury problems". Their solutions are not wave-like . The solution at any point *interior* to the domain depends on the conditions specified on the entire boundary of the closed domain . Solutions are generally very smooth 
  • Canonical Form: Can be reduced to the form φξξ + φηη = h(...).
  • Examples:
    • Laplace's equation: ∂2U/∂x2 + ∂2U/∂y2 = 0 . This governs steady-state temperature distribution in solids and incompressible irrotational fluid flow .
    • Poisson's equation: ∂2U/∂x2 + ∂2U/∂y2 = f(x, y) . This is similar to Laplace's equation but includes a source term .
  • Well-Posed Problems: For elliptic equations, the problem requires a closed domain and specific types of boundary conditions :
    • Dirichlet problem: The function u itself is prescribed on the boundary .
    • Neumann problem: The normal derivative of u (∂u/∂n) is specified on the boundary .
    • Mixed or Robin's problem: A combination of the function and its normal derivative is specified on the boundary .
    An elliptic problem on an open domain with boundary conditions only on part of the boundary can be ill-posed .

5. Classification of Systems of Equations

Classifying systems of first-order PDEs can be more complex than for a single second-order PDE [1]. However, similar principles apply.

  • For a linear system of first-order equations, Ut + [A]Ux = 0, the system's type (hyperbolic, parabolic, or elliptic) is determined by the eigenvalues (λ) of the matrix [A] .
    • If all eigenvalues are real, the system is hyperbolic.
    • If there is a single real root, it can be parabolic .
    • If the eigenvalues are complex, the system is elliptic .
  • For example, the system of equations representing the wave equation can be classified as hyperbolic based on the real eigenvalues of its associated matrix [A] .
  • The classification of systems of *second-order* PDEs is very complex, and their mathematical behavior can be difficult to determine except in simple cases . This includes the governing equations of fluid dynamics, which are a coupled system of nonlinear partial differential equations.