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On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. (English) Zbl 0880.65076
The paper deals with the computation of numerical solutions for the nonlinear one-dimensional evolution equation $u_t=\cal Lu+ \cal Nf(u)$, with initial condition $u(x,0)=u(x)$, $0\leq x \leq 1$, and the periodic boundary condition $u(0,t)=u(1,t)$, $0\leq t \leq T$. $\cal L$ and $\cal N$ are constant differential operators that do not depend on time, whereas $f$ usually is a nonlinear function. With appropriate choices of the differential operators, the equation can turn to be Burgers’ equation, its generalised version, or Burgers’ cubic equation. The proposed new approach has as its main component the observed fact that wide classes of operators can have sparse representations in wavelet bases, thus yielding fast algorithms for applying these operators to functions and for evaluating the pointwise product of functions represented in those bases. In the case at hand, the original equation is first transformed into the integral equation formulation given by semigroup theory. The integral is then approximated by a convenient quadrature in operator function; this approximation is dealt with by precomputing the nonstandard form of the operator functions, i.e. their wavelet expressions, and then applying them as necessary. The actual computations are carried on by means of the two algorithms already mentioned; the first one applies adaptively operators to functions developed in the wavelet basis, i.e. by taking into account only the wavelet function coefficients whose magnitude is above a given threshold. The second algorithm evaluates, adaptively as well, the pointwise product of functions represented in wavelet bases; this algorithm can be extended to obtain the evaluation of $f(u)$, with $f$ an analytic function with rapidly convergent Taylor series. Both algorithms have an operation count that is proportional to the number of significant wavelet coefficients in the expansion of the involved functions. In the numerical experiments section, the paper examines how the approach proposed works on the heat equation and the three Burgers variants mentioned above. For the former, the unconditionally stable Crank-Nicolson scheme is chosen, namely $AU(t_{j+1})=BU(t_j)$, where $U(t_{j+1})$ are the values to be calculated at time level $t_j$, and $A$ and $B$ are the scheme’s tridiagonal matrices. The comparison of the nonstandard form of $A^{-1}B$ with that of the exponential of the second derivative shows that the latter has a much sparser structure than the former, which is a very good justification for applying the wavelet approach to the semigroup formulation. The rest of the section examines in all detail the consequences of the application of the proposed algorithms, such as the error committed and how the computational work evolves with time. The paper demonstrates abundantly that this new approach, for the case examined at least, combines all the good features of the approaches usually applied to nonlinear partial differential equations. Further extensions are to follow.

##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M06 Finite difference methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 35K55 Nonlinear parabolic equations 35Q53 KdV-like (Korteweg-de Vries) equations
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