520 Difference Methods for Solving Nonlinear Equations- Various implicit schemes for the quasilinear heat conduction equation.
Other ideas are connected with two types of purely implicit difference
schemes (the forward ones with (J = 1) available for the simplest quasi-
linear heat conduction equation
(33) OU ot = ox^0 ( k(u) 01l) ox + f(u)'. O<x<l, O<t<T,
u(x, 0) = u 0 (x), u(O, t) = U 1 (t), u(l, t) = u 2 (t),
where k(u) > 0.
The structures of both schemes are well-characterized byfor the scheme a) and by( 35 ) Yi -T Yi _ - h 1 [ ai+l (·') y Yi+1 h - Yi - a; (')Yi y -Yi-1] h + f(') Yi
for the scheme b), where Yi = yf ·+1 , Yi yf, a; ( v)
example, we might agree to consider(36) a;(v) = 0.5 (k(vi-I + k(u;)),
(38)A greater gain in accuracy in connection with the temperature wave
depends significantly on how well we calculate the coefficients a; ( v). In
the case where k = k 0 u" is a power function of temperature, numerical
experin1ents showed that fon11ula (38) is useless and formula (36) is n1uch
rnore flexible than (37), so there is son1e reason to be concerned about this.
Further comparison of schemes (34) and (35) should cause some difficulties.
Both schemes are absolutely stable and have the same error of approxima-
tion 0( r + h^2 ). The scheme a) is linear with respect to the value of the
function yj+l on the layer tj+l and so the value yj+i on every new layer