D03 (Pde)

Partial Differential Equations

This chapter is concerned with the numerical solution of partial differential equations.
Currently only solvers for parabolic and hyperbolic equations are included.

The definition of a partial differential equation problem includes not only the equation itself but also the domain of interest and appropriate subsidiary conditions. Indeed, partial differential equations are usually classified as elliptic, hyperbolic or parabolic according to the form of the equation **and** the form of the subsidiary conditions which must be assigned to produce a well-posed problem. The functions in this chapter will often call upon functions from other chapters, such as Chapter F04 (Simultaneous Linear Equations) and Chapter D02 (Ordinary Differential Equations). Other chapters also contain relevant functions, in particular Chapter D06 (Mesh Generation) and Chapter F11 (Large Scale Linear Systems).

The classification of partial differential equations is easily described in the case of **linear** equations of the **second order** in two independent variables, i.e., equations of the form

where $a$, $b$, $c$, $d$, $e$, $f$ and $g$ are functions of $x$ and $y$ only. Equation (1) is called elliptic, hyperbolic or parabolic according to whether $ac-{b}^{2}$ is positive, negative or zero, respectively. Useful definitions of the concepts of elliptic, hyperbolic and parabolic character can also be given for differential equations in more than two independent variables, for systems and for nonlinear differential equations.

$$a{u}_{xx}+2b{u}_{xy}+c{u}_{yy}+d{u}_{x}+e{u}_{y}+fu+g=0\text{,}$$ | (1) |

For **elliptic** equations, of which Laplace's equation

is the simplest example of second order, the subsidiary conditions take the form of **boundary** conditions, i.e., conditions which provide information about the solution at all points of a **closed** boundary. For example, if equation (2) holds in a plane domain $D$ bounded by a contour $C$, a solution $u$ may be sought subject to the condition

where $f$ is a given function. The condition (3) is known as a Dirichlet boundary condition. Equally common is the Neumann boundary condition

which is one form of a more general condition

where ${u}^{\prime}$ denotes the derivative of $u$ normal to the contour $C$, and $f$ and $g$ are given functions. Provided that $f$ and $g$ satisfy certain restrictions, condition (5) yields a well-posed **boundary value problem** for Laplace's equation. In the case of the Neumann problem, one further piece of information, e.g., the value of $u$ at a particular point, is necessary for uniqueness of the solution. Boundary conditions similar to the above are applicable to more general second-order elliptic equations, whilst two such conditions are required for equations of fourth order.

$${u}_{xx}+{u}_{yy}=0$$ | (2) |

$$u=f\text{\hspace{1em} on \hspace{1em}}C\text{,}$$ | (3) |

$${u}^{\prime}=g\text{\hspace{1em} on \hspace{1em}}C\text{,}$$ | (4) |

$${u}^{\prime}+fu=g\text{\hspace{1em} on \hspace{1em}}C\text{,}$$ | (5) |

For **hyperbolic** equations, the wave equation

is the simplest example of second order. It is equivalent to a first-order system

The subsidiary conditions may take the form of **initial** conditions, i.e., conditions which provide information about the solution at points on a suitable **open** boundary. For example, if equation (6) is satisfied for $t>0$, a solution $u$ may be sought such that

where $f$ and $g$ are given functions. This is an example of an **initial value problem**, sometimes known as Cauchy's problem.

$${u}_{tt}-{u}_{xx}=0$$ | (6) |

$${u}_{t}-{v}_{x}=0\text{, \hspace{1em}}{v}_{t}-{u}_{x}=0\text{.}$$ | (7) |

$$u(x,0)=f\left(x\right),\text{\hspace{1em}}{u}_{t}(x,0)=g\left(x\right)\text{,}$$ | (8) |

For **parabolic** equations, of which the heat conduction equation

is the simplest example, the subsidiary conditions always include some of **initial** type and may also include some of **boundary** type. For example, if equation (9) is satisfied for $t>0$ and $0<x<1$, a solution $u$ may be sought such that

and

This is an example of a mixed **initial/boundary value problem**.

$${u}_{t}-{u}_{xx}=0$$ | (9) |

$$u(x,0)=f\left(x\right),\text{\hspace{1em}}0<x<1\text{,}$$ | (10) |

$$u(0,t)=0,\text{\hspace{1em}}u(1,t)=1,\text{\hspace{1em}}t>0\text{.}$$ | (11) |

For all types of partial differential equations, finite difference methods (see Mitchell and Griffiths (1980)) and finite element methods (see Wait and Mitchell (1985)) are the most common means of
solution and such methods obviously feature prominently in this chapter.
Some of the utility functions in this chapter are concerned with the solution of the large sparse systems of equations which arise from finite difference and finite element methods. Further functions for this purpose are provided in Chapter F11.

Alternative methods of solution are often suitable for special classes of problems. For example, the method of characteristics is the most common for hyperbolic equations involving time and one space dimension (see Smith (1985)). The method of lines (see Mikhlin and Smolitsky (1967)) may be used to reduce a parabolic equation to a (stiff) system of ordinary differential equations, which may be solved by means of functions from Chapter D02 (Ordinary Differential Equations). Similarly, integral equation or boundary element methods (see Jaswon and Symm (1977)) are frequently used for elliptic equations. Typically, in the latter case, the solution of a boundary value problem is represented in terms of certain boundary functions by an integral expression which satisfies the differential equation throughout the relevant domain. The boundary functions are obtained by applying the given boundary conditions to this representation. Implementation of this method necessitates discretization of only the boundary of the domain, the dimensionality of the problem thus being effectively reduced by one. The boundary conditions yield a full system of simultaneous equations, as opposed to the sparse systems yielded by finite difference and finite element methods, but the full system is usually of much lower order. Solution of this system yields the boundary functions, from which the solution of the problem may be obtained, by quadrature, as and where required.

See Section 3.8.

There are five functions available for solving general parabolic equations in one space dimension:

Equations may include nonlinear terms but the true derivative ${u}_{t}$ should occur linearly and equations should usually contain a second-order space derivative ${u}_{xx}$. There are certain restrictions on the coefficients to try to ensure that the problems posed can be solved by the above functions.

The method of solution is to discretize the space derivatives using finite differences or collocation, and to solve the resulting system of ordinary differential equations using a ‘stiff’ solver.

d03pcc and d03pdc can solve a system of parabolic equations of the form

where $i=1,2,\dots ,n$, $a\le x\le b$, $t\ge {t}_{0}$.

$$\sum _{j=1}^{n}{P}_{ij}(x,t,U,{U}_{x})\frac{\partial {U}_{j}}{\partial t}+{Q}_{i}(x,t,U,{U}_{x})={x}^{-m}\frac{\partial}{\partial x}\left({x}^{m}{R}_{i}(x,t,U,{U}_{x})\right)\text{,}$$ |

The parameter $m$ allows the function to handle different coordinate systems easily (Cartesian, cylindrical polars and spherical polars). d03pcc uses a finite differences spatial discretization and d03pdc uses a collocation spatial discretization.

d03phc and d03pjc are similar to d03pcc and d03pdc respectively, except that they provide scope for coupled differential-algebraic systems. This extended functionality allows for the solution of more complex and more general problems, e.g., periodic boundary conditions and integro-differential equations.

d03ppc is similar to d03phc but allows remeshing to take place in the spatial direction. This facility can be very useful when the nature of the solution in the spatial direction varies considerably over time.

d03ncc solves the Black–Scholes equation

for the value $f$ of a European or American, put or call stock option. The parameters $r$, $q$ and $\sigma $ may each be either constant or time-dependent. The values of the Greeks are also returned.

$$\frac{\partial f}{\partial t}+(r-q)S\frac{\partial f}{\partial S}+\frac{{\sigma}^{2}{S}^{2}}{2}\frac{{\partial}^{2}f}{\partial {S}^{2}}=rf$$ |

$${S}_{\mathrm{min}}<S<{S}_{\mathrm{max}}\text{, \hspace{1em}}{t}_{\mathrm{min}}<t<{t}_{\mathrm{max}}\text{,}$$ |

In certain cases an analytic solution of the Black–Scholes equation is available. In these cases the solution may be computed by d03ndc. More generally, Chapter S contains a set of option pricing functions that evaluate the closed form solutions or approximations to the equations that define mathematical models for the prices of selected financial option contracts, including the Black–Scholes equation (s30aac).

There are three functions available for solving systems of first-order partial differential equations:

Equations may include nonlinear terms but the time derivative should occur linearly. There are certain restrictions on the coefficients to ensure that the problems posed can be solved by the above functions.

The method of solution is to discretize the space derivatives using the Keller box scheme and to solve the resulting system of ordinary differential equations using a ‘stiff’ solver.

d03pec is designed to solve a system of the form

where $i=1,2,\dots ,n$, $a\le x\le b$, $t\ge {t}_{0}$.

$$\sum _{j=1}^{n}{P}_{ij}(x,t,U,{U}_{x})\frac{\partial {U}_{j}}{\partial t}+{Q}_{i}(x,t,U,{U}_{x})=0\text{,}$$ |

d03pkc is similar to d03pec except that it provides scope for coupled differential algebraic systems. This extended functionality allows for the solution of more complex problems.

d03prc is similar to d03pkc but allows remeshing to take place in the spatial direction. This facility can be very useful when the nature of the solution in the spatial direction varies considerably over time.

d03pec, d03pkc or d03prc may also be used to solve systems of higher or mixed order partial differential equations which have been reduced to first-order. Note that in general these functions are unsuitable for hyperbolic first-order equations, for which an appropriate upwind discretization scheme should be used (see Section 3.8 for example).

There are three functions available for solving systems of convection-diffusion equations with optional source terms:

Equations may include nonlinear terms but the time derivative should occur linearly. There are certain restrictions on the coefficients to ensure that the problems posed can be solved by the above functions, in particular the system must be posed in conservative form (see below). The functions may also be used to solve hyperbolic convection-only systems.

Convection terms are discretized using an upwind scheme involving a numerical flux function based on the solution of a Riemann problem at each mesh point (see LeVeque (1990)); and diffusion and source terms are discretized using central differences. The resulting system of ordinary differential equations is solved using a ‘stiff’ solver. In the case of Euler equations for a perfect gas various approximate and exact Riemann solvers are provided in d03puc, d03pvc, d03pwc and d03pxc. These functions may be used in conjunction with d03pfc, d03plc and d03psc.

d03pfc is designed to solve systems of the form

$$\sum _{j=1}^{n}{P}_{ij}(x,t,U)\frac{\partial {U}_{j}}{\partial t}+\frac{\partial}{\partial x}{F}_{i}(x,t,U)={C}_{i}(x,t,U)\frac{\partial}{\partial x}{D}_{i}(x,t,U,{U}_{x})+{S}_{i}(x,t,U)\text{,}$$ |

$$\sum _{j=1}^{n}{P}_{ij}(x,t,U)\frac{\partial {U}_{j}}{\partial t}+\frac{\partial {F}_{i}(x,t,U)}{\partial x}=0\text{,}$$ |

d03plc is similar to d03pfc except that it provides scope for coupled differential algebraic systems. This extended functionality allows for the solution of more complex problems.

d03psc is similar to d03plc but allows remeshing to take place in the spatial direction. This facility can be very useful when the nature of the solution in the spatial direction varies considerably over time.

A range of mesh generation functions are available in Chapter D06.

Functions are available in the Linear Algebra Chapters for the direct and iterative solution of linear equations. Here we point to some of the functions that may be of use in solving the linear systems that arise from finite difference or finite element approximations to partial differential equation solutions. Chapters F01, F04, F07, F08 and F11 should be consulted for further information and for the appropriate function documents. Decision trees for the solution of linear systems are given in Section 4 in the F04 Chapter Introduction.

The following functions allow the direct solution of symmetric positive definite systems:

and the following functions allow the iterative solution of symmetric positive definite and symmetric-indefinite systems:

Band | f07hdc and f07hec |

Variable band (skyline) | f01mcc and f04mcc |

Sparse | f11jac, f11jcc and f11jec. |

The latter two functions above are Black Box functions which include Incomplete Cholesky, SSOR or Jacobi preconditioning.

The following functions allow the direct solution of nonsymmetric systems:

and the following functions allow the iterative solution of nonsymmetric systems:

Band | f07bdc and f07bec |

Sparse | f11dac, f11dcc and f11dec. |

The latter two functions above are Black Box functions which include incomplete $LU$, SSOR and Jacobi preconditioning.

The functions d03pyc and d03pzc use linear interpolation to compute the solution to a parabolic problem and its first derivative at the user-specified points. d03pzc may be used in conjunction with d03pcc, d03pec, d03phc, d03pkc, d03ppc and d03prc. d03pyc may be used in conjunction with d03pdc and d03pjc.

Does PDE have a time derivative? | Does PDE have second derivatives? | see Tree 3 Parabolic branch | |||||||||

yes | yes | ||||||||||

no | no | ||||||||||

Is PDE hyperbolic? | see Tree 2 Hyperbolic branch | ||||||||||

yes | |||||||||||

no | |||||||||||

1 space dimension? | Does PDE have coupled ODEs? | Is a remeshing process required? | d03prc | ||||||||

yes | yes | yes | |||||||||

no | no | no | |||||||||

d03pkc | |||||||||||

d03pec | |||||||||||

N/A | |||||||||||

No elliptic solvers currently available. | |||||||||||

1 space dimension? | Does PDE have coupled ODEs? | Is a remeshing process required? | d03psc | |||||||

yes | yes | yes | ||||||||

no | no | no | ||||||||

d03plc | ||||||||||

d03pfc | ||||||||||

N/A | ||||||||||

1 space dimension? | Is PDE the Black–Scholes equations? | d03ncc or d03ndc | |||||||||

yes | yes | ||||||||||

no | no | ||||||||||

Is PDE in conservative form? | Does PDE have coupled ODEs? | Is a remeshing process required? | d03psc | ||||||||

yes | yes | yes | |||||||||

no | no | no | |||||||||

d03plc | |||||||||||

d03pfc | |||||||||||

see Tree 4 Branch for parabolic PDE in non-conservative form | |||||||||||

N/A | |||||||||||

Do you want to use finite differences? | Does PDE have coupled ODEs? | Is a remeshing process required? | d03ppc | |||||||

yes | yes | yes | ||||||||

no | no | no | ||||||||

d03phc | ||||||||||

d03pcc | ||||||||||

Do you want to use Chebyshev collocation? | Does PDE have coupled ODEs? | d03pjc | ||||||||

yes | yes | |||||||||

no | no | |||||||||

d03pdc | ||||||||||

N/A | ||||||||||

Black–Scholes equation, |

analytic | d03ndc |

finite difference | d03ncc |

Convection-diffusion system(s), |

nonlinear, |

one space dimension, |

using upwind difference scheme based on Riemann solvers | d03pfc |

using upwind difference scheme based on Riemann solvers, |

with coupled differential algebraic system | d03plc |

with remeshing | d03psc |

First-order system(s), |

nonlinear, |

one space dimension, |

using Keller box scheme | d03pec |

using Keller box scheme, |

with coupled differential algebraic system | d03pkc |

with remeshing | d03prc |

PDEs, general system, one space variable, method of lines, |

parabolic, |

collocation spatial discretization, |

coupled DAEs, comprehensive | d03pjc |

easy-to-use | d03pdc |

finite differences spatial discretization, |

coupled DAEs, comprehensive | d03phc |

coupled DAEs, remeshing, comprehensive | d03ppc |

easy-to-use | d03pcc |

Utility function, |

average values for d03ndc | d03nec |

exact Riemann solver for Euler equations | d03pxc |

HLL Riemann solver for Euler equations | d03pwc |

interpolation function for collocation scheme | d03pyc |

interpolation function for finite difference, |

Keller box and upwind scheme | d03pzc |

Osher's Riemann solver for Euler equations | d03pvc |

Roe's Riemann solver for Euler equations | d03puc |

None.

None.

Ames W F (1977) *Nonlinear Partial Differential Equations in Engineering* (2nd Edition) Academic Press

Berzins M (1990) Developments in the NAG Library software for parabolic equations *Scientific Software Systems* (eds J C Mason and M G Cox) 59–72 Chapman and Hall

Jaswon M A and Symm G T (1977) *Integral Equation Methods in Potential Theory and Elastostatics* Academic Press

LeVeque R J (1990) *Numerical Methods for Conservation Laws* Birkhäuser Verlag

Mikhlin S G and Smolitsky K L (1967) *Approximate Methods for the Solution of Differential and Integral Equations* Elsevier

Mitchell A R and Griffiths D F (1980) *The Finite Difference Method in Partial Differential Equations* Wiley

Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations *ACM Trans. Math. Softw.* **20** 63–99

Richtmyer R D and Morton K W (1967) *Difference Methods for Initial-value Problems* (2nd Edition) Interscience

Smith G D (1985) *Numerical Solution of Partial Differential Equations: Finite Difference Methods* (3rd Edition) Oxford University Press

Swarztrauber P N and Sweet R A (1979) Efficient Fortran subprograms for the solution of separable elliptic partial differential equations *ACM Trans. Math. Software* **5** 352–364

Wait R and Mitchell A R (1985) *Finite Element Analysis and Application* Wiley