Explicit Method for the General Non-linear Parabolic Problem

Consider the problem defined for an arbitrary function Φ,

u_{t} = Φ(x, t, u, u_{x}, u_{xx})  in 0 < x < 1 = Nh,  0 < t < T = Jk  (7.3)

For physical sense, we will assume that the effective diffusivity is positive and bounded, so that

A ⩾ \frac{∂Φ}{∂u_{xx}} ⩾ a > 0

for some constants A and a. A simple, centred, explicit method replaces

\begin{matrix}u_{x}  by \frac{1}{2h}△U_{n}^{j} & ≡ \frac{U_{n+1}^{j} − U_{n−1}^{j}}{2h} \\ u_{xx}  by \frac{1}{h^{2}} δ^{2}U_{n}^{j} & ≡ \frac{U_{n+1}^{j} + U_{n−1}^{j} − 2U_{n}^{j}}{h^{2}} \end{matrix} \} + O(h^{2})  (7.4)

so that

U_{n}^{j+1} = U_{n}^{j} + kΦ[nh, jk, U_{n}^{j}, \frac{△U_{n}^{j}}{2h} , \frac{δ^{2}U_{n}^{j}}{h^{2}} ]  (7.5)

We define the local error z_{n}^{j} = u_{n}^{j} − U_{n}^{j} and investigate the conditions for which we can demonstrate stability using a Maximum Principle Analysis (MPA). Now u obeys the same equation as U with a truncation error R_{n}^{j} = O(k, h^{2}) added in. Furthermore,

\begin{aligned}Φ[nh, jk, u_{n}^{j}, \frac{△u_{n}^{j}}{2h} , \frac{δ^{2}u_{n}^{j}}{h^{2}} ] & − Φ[nh, jk, U_{n}^{j}, \frac{△U_{n}^{j}}{2h} , \frac{δ^{2}U_{n}^{j}}{h^{2}} ] \\ & = \frac{∂Φ}{∂u}z_{n}^{j} + \frac{∂Φ}{∂u_{x}} \frac{△z_{n}^{j}}{2h} + \frac{∂Φ}{∂u_{xx}} \frac{δ^{2}z_{n}^{j}}{h^{2}} + O((z)^{2}) . \end{aligned}

Subtracting (7.5) from the equation involving u and using the above, gives

z_{n}^{j+1} = r[\frac{∂Φ}{∂u_{xx}} − \frac{h}{2} \frac{∂Φ}{∂u_{x}} ]z_{n−1}^{j} + r[\frac{∂Φ}{∂u_{xx}} + \frac{h}{2} \frac{∂Φ}{∂u_{x}} ]z_{n+1}^{j} + [1 + k \frac{∂Φ}{∂u} − 2r \frac{∂Φ}{∂u_{xx}} ]z_{n}^{j} + kR_{n}^{j}

(7.6)

Now the Maximum Principle argument requires the three coefficients in square brackets to be positive (see the arguments of §3, equation (3.4) etc.). Suppose therefore that

A ⩾ \frac{∂Φ}{∂u_{xx}} ⩾ a > 0,  |\frac{∂Φ}{∂u_{x}} | ⩽ b ,  C ⩾ \frac{∂Φ}{∂u} ⩾ c ,  (7.7)

where b > 0, c and C are constants. Then

\frac{∂Φ}{∂u_{xx}} ± \frac{1}{2}h\frac{∂Φ}{∂u_{x}} ⩾ a − \frac{1}{2}hb  and 1 + k \frac{∂Φ}{∂u} − 2r \frac{∂Φ}{∂u_{xx}} ⩾ 1 + kc − 2rA  (7.8)

For the MPA, we therefore require

a − \frac{1}{2}hb ⩾ 0  and 1 + kc − 2rA ⩾ 0  (7.9)

If (7.9) holds then (7.6) implies

|z_{n}^{j+1}| ⩽ (1 + k \frac{∂Φ}{∂u}) max_{n} [ |z_{n}^{j}| ] + kR_{n}^{j} ⩽ (1 + kC) ||z^{j}|| + D(k^{2} + kh^{2})

for some constant D > 0, and so

||z^{j+1}|| ⩽ (1 + kC) ||z^{j}|| + D(k^{2} + kh^{2})

Now the initial error ||z^{0}|| = 0. If C > 0 we get bounded growth (this is the |λ| ⩽ 1+O(k) case). It can be shown from the above that

||z^{j}|| ⩽ \frac{e^{Cjk} − 1}{C}D(k + h^{2})  for 1 ⩽ j ⩽ J  (7.10)

By choosing k and h small enough for fixed T = Jk we can ensure that the errors are as small as we choose. (Note that the exponential growth in the error term is due to the p.d.e. itself possessing exponentially growing solutions. Compare the equation u_{t} = cu.) We conclude that the explicit method will work for the nonlinear equation (7.3) provided (7.9) holds.

As an example, consider Burger’s equation

u_{t} + uu_{x} = νu_{xx}  so that  Φ = νu_{xx} − uu_{x} ,

where ν is constant. Then

A = a = ν,  b = max_{x,t} [|u|] ,  c = min_{x,t} [−u_{x}] , C = max_{x,t} [−u_{x}] .

So the explicit method for this equation is stable if

ν − \frac{1}{2}h max [|u|] ⩾ 0  or  |u| ⩽ \frac{2ν}{h}
and 1 + k min [ − u_{x}] − 2rν ⩾ 0  or  − u_{x} ⩾ \frac{2rν − 1}{k} .

The first of these conditions is a restriction on the size of the spatial steplength h for large values of the advective velocity u. In the absence of diffusion (ν = 0) the centred scheme given by (7.4) is always unstable (see 10.7A, later). The second condition is a generalisation of the ‘r < 1/2’ relation with which we are familiar. We note that if 2rν > 1, no value of k will guarantee a stable scheme, and as k → 0 the stability condition is violated.