Applied Numerical Mathematics 7 (1991) 287-296 North-Holland 287 Sum-accelerated pseudospectral methods: the Euler-accelerated sine algorithm zyxwvutsrqponmlkjihgfedc John P. Boyd Department of Atmospheric, Oceanic and Space Science and Luboratoty for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, USA Abstract Boyd, J.P., Sum-accelerated pseudospectral methods: the Euler-accelerated sine algorithm, Applied Numerical Mathematics 7 (1991) 287-296 Pseudospectral discretizations of differential equations are much more accurate than finite differences for the same number of grid points N. The reason is that derivatives are approximated by a weighted sum of all N values of u(x,), rather than just three as in a second-order finite difference. The price is that the N zyxwvutsrqp X N pseudospectral matrix is dense with N nonzero elements (rather than three) in each row. Truncating the pseudospectral sums to create a sparse discretization fails because the derivative series are alternating and very slowly convergent. However, these series are perfect candidates for sum-acceleration methods. We show that the Euler summation can be applied to a standard pseudospectral scheme to produce an algorithm which is both exponentially accurate (like any other spectral method) and yet generates sparse matrices (like a finite difference method). For illustration, we use the sine basis with an evenly spaced grid on x E [ - co, 001. However, the same techniques apply equally well to Chebyshev and Fourier polynomials. zyxwvutsrqponm 1. Introduction Finite difference methods and pseudospectral schemes both approximate a derivative by a weighted sum of the values of U(X), i.e., du dx x=x = 4dxd + c {Ak+i+d + A-&-d)- k=l 0.1) For a second-order centered finite difference, for example, the sum is truncated at k = 1 and the weights are A,, = 0 and A + I = + 1/ (2/ z). For pseudospectral methods, however, the sum extends over all points on the grid. For the particular case of the sine basis, the grid is evenly spaced and extends over the whole interval x E [ - 00, co]; equation (1.1) is an infinite series. The reward for using more terms is more accuracy: a pseudospectral method with N points has an error that decreases exponentia& fast as N increases, in contrast to the 0(1/ N*) error of a second-order difference. The rub is that when such approximations are used to discretize a differential equation and thus convert it into a matrix problem, the pseudospectral matrix is dense whereas the corresponding finite difference matrix is sparse. An obvious question is: Can one somehow combine the good features of both spectral and finite difference methods by inventing an algorithm that is both sparse and exponentially accurate? 016%9274/ 91/ $03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)