Solves a system of partial differential equations of the form ut = f(x, t, u, ux, uxx) using the method of lines. The solution is represented with cubic Hermite polynomials.

Let M = NPDES, N = NX and xi = XBREAK(I). The routine MMOLCH uses the method of lines to solve the partial differential equation system

with the initial conditions

and the boundary conditions

for k = 1, …, M.

Cubic Hermite polynomials are used in the x variable approximation so that the trial solution is expanded in the series

where ɸi(x) and Ψi(x) are the standard basis functions for the cubic Hermite polynomials with the knots x1 < x2 < … < xN. These are piecewise cubic polynomials with continuous first derivatives. At the breakpoints, they satisfy

According to the collocation method, the coefficients of the approximation are obtained so that the trial solution satisfies the differential equations at the two Gaussian points in each subinterval,

for j = 1, …, N. The collocation approximation to the differential equation is

for k = 1, …, M and j = 1, …, 2(N − 1).

If αk = βk = 0, it is assumed that no boundary condition is desired for the k-th unknown at the left endpoint. A similar comment holds for the right endpoint. Thus, collocation is done at the endpoint. This is generally a useful feature for systems of first-order partial differential equations.

The input/output array Y contains the values of the ai,k. The initial values of the bi,k are obtained by using the IMSL cubic spline routine CSINT (see Chapter 3, “Interpolation and Approximation”) to construct functions

such that

The IMSL routine CSDER, (see Chapter 3, “Interpolation and Approximation”), is used to approximate the values

If INPDER = 1, the user should provide the initial values of bi,k.

The order of matrix A is 2MN and its maximum bandwidth is 6M-1. The band structure of the Jacobian of F with respect to c is the same as the band structure of A. This system is solved using a modified version of IVPAG. Numerical Jacobians are used exclusively. The algorithm is unchanged. Gear’s BDF method is used as the default because the system is typically stiff. For more details, see Sewell (1982).

We now present three examples of PDEs that illustrate how users can interface their problems with IMSL PDE solving software. The examples are small and not indicative of the complexities that most practitioners will face in their applications. A set of seven sample application problems, some of them with more than one equation, is given in Sincovec and Madsen (1975). Two further examples are given in Madsen and Sincovec (1979).

The normalized linear diffusion PDE, ut = uxx, 0 ≤ x ≤ 1, t > 0, is solved. The initial values are u(x, 0) = u0 = 1. There is a “zero-flux” boundary condition at x = 1, namely ux(1, t) = 0, (t > 0). The boundary value of u(0, t) is abruptly changed from u0 to the value 0, for t > 0.

When the boundary conditions are discontinuous, or incompatible with the initial conditions such as in this example, it may be important to use double precision.

In this example, using MMOLCH, we solve the linear normalized diffusion PDE ut = uxx but with an optional usage that provides values of the derivatives, ux, of the initial data. Due to errors in the numerical derivatives computed by spline interpolation, more precise derivative values are required when the initial data is u(x, 0) = 1 + cos[(2n-1)πx], n > 1. The boundary conditions are “zero flux” conditions ux(0, t) = ux(1, t) = 0 for t > 0.

In this example, we consider the linear normalized hyperbolic PDE, utt = uxx, the “vibrating string” equation. This naturally leads to a system of first order PDEs. Define a new dependent variable ut = v. Then, vt = uxx is the second equation in the system. We take as initial data u(x,0) = sin(πx) and ut(x, 0) = v(x, 0) = 0. The ends of the string are fixed so u(0, t) = u(1, t) = v(0, t) = v(1, t) = 0. The exact solution to this problem is u(x, t) = sin(πx) cos(πt). Residuals are computed at the output values of t for 0 < t≤2. Output is obtained at 200 steps in increments of 0.01.

Even though the sample code MMOLCH gives satisfactory results for this PDE, users should be aware that for nonlinear problems, “shocks” can develop in the solution. The appearance of shocks may cause the code to fail in unpredictable ways. See Courant and Hilbert (1962), pages 488-490, for an introductory discussion of shocks in hyperbolic systems.