Numerical solutions of multi-dimensional partial differential equations using an adaptive wavelet method (original) (raw)

On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases

Journal of Computational Physics, 1997

This work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form u t ϭ Any wavelet-expansion approach to solving differential L u ϩ N f (u), where L and N are linear differential operators and equations is essentially a projection method. In a projection f (u) is a nonlinear function. These equations are adaptively solved method the goal is to use the fewest number of expansion by projecting the solution u and the operators L and N into a coefficients to represent the solution since this leads to wavelet basis. Vanishing moments of the basis functions permit a sparse representation of the solution and operators. Using these efficient numerical computations. The number of coeffisparse representations fast and adaptive algorithms that apply opercients required to represent a function expanded in a Fouators to functions and evaluate nonlinear functions, are developed rier series (or similar expansions based on the eigenfuncfor solving evolution equations. For a wavelet representation of the tions of a differential operator) depends on the most solution u that contains N s significant coefficients, the algorithms singular behavior of the function. We are interested in update the solution using O(N s) operations. The approach is applied to a number of examples and numerical results are given. ᮊ 1997 solutions of partial differential equations that have regions Academic Press of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shock-like structures. Therefore, expansions of these solutions, based upon * Research partially supported by ONR Grant N00014-91-J4037 and Calderó n-Zygmund and pseudo-differential operators, ARPA Grant F49620-93-1-0474.

Adaptive Wavelet Methods II—Beyond the Elliptic Case

Foundations of Computational Mathematics, 2002

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in ) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve Date A. Cohen, W. Dahmen, and R. DeVore the resulting infinite system of equations. This second step requires a significant modification of the ideas from . The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.

A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs

Journal of Computational Physics, 1997

A fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed. The computational cost of the algorithm is independent of the dimensionality of the problem and is O(N ), where N is the total number of collocation points. The method can handle general boundary conditions. The multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution. High resolution computations are performed only in regions where singularities or sharp transitions occur. Numerical results demonstrate the ability of the method to resolve localized structures such as shocks, which change their location and steepness in space and time. The present results indicate that the method has clear advantages in comparison with well established numerical algorithms. ᮊ 1997 Academic Press

A Wavelet Collocation Method for Solving PDEs

2001

Abstract This report provides an overview of a recent paper by Vasilyev and Bowman [J. Comp. Phys., 165: 660–693, 2000]. The paper discusses the use of so-called second-generation wavelet bases in a method-of-lines approach to the numerical solution of partial differential equations. The focus of this review is on the paper's justification of its adaptive grid method based on the approximating properties of a wavelet basis.

Second-Generation Wavelet Collocation Method for the Solution of Partial Differential Equations

Journal of Computational Physics, 2000

An adaptive numerical method for solving partial differential equations is developed. The method is based on the whole new class of second-generation wavelets. Wavelet decomposition is used for grid adaptation and interpolation, while a new O(N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm. In this paper we demonstrate the algorithm for one particular choice of second-generation wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using second-generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling. Thus, the strength of the new method is that it can be easily extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose the wavelet basis depending on the application.

Wavelets for Partial Differential Equations

The property of wavelet characterization of Sobolev and Besov spaces that we saw in the previous section is quite a powerful tool. In the next sections we will see how we can take advantage of such a property in the design of new efficient methods for the solution of PDEs. Let us assume from now on that we have a couple of multiresolution analyses V j andṼ j satisfying all space and frequency localization assumptions of Section 2.2. 3.1 Wavelet Preconditioning The first example of application of wavelets to the numerical solution of PDEs is the construction of optimal preconditioners for elliptic partial differential and pseudo differential equations [30, 23].

A Wavelet Based Numerical Method for Nonlinear Partial Differential Equations

e-pub.uni-weimar.de

This paper is concerned with the numerical treatment of quasilinear elliptic partial differential equations. In order to solve the given equation we propose to use a Galerkin approach, but, in contrast to conventional finite element discretizations, we work with trial spaces that, not only ...