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

Direct Solution of Differential Equations Using a Wavelet-Based Multiresolution Method

2011

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelets as basis functions in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties, especially for problems with local high gradient, which would require a dense mesh in tradicional methods, like the FEM. Another possible advantage is the fact that the calculation of the integrals of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. In this work, the Galerkin Method has been adapted for the direct solution of diff...

Numerical Solution of Partial Differential Equations Using Wavelet Approximation Space

2000

Powell-Sabin splines are piecewise quadratic polynomials with global C 1 -continuity. They are defined on conformal triangulations of two-dimensional domains, and admit a compact representation in a normalized B-spline basis. Recently, these splines have been used successfully in the area of computer-aided geometric design for the modelling and fitting of surfaces. In this paper, we discuss the applicability of Powell-Sabin splines for the numerical solution of partial differential equations defined on irregular domains. A Galerkin-type PDE discretisation is derived, and elaborated for the variable coefficient diffusion equation. Special emphasis goes to the treatment of Dirichlet and Neumann boundary conditions. Finally, an error estimator is developed and an adaptive mesh refinement strategy is proposed. We will illustrate the effectiveness of the approach by means of some numerical experiments.

2-D wavelet-based adaptive-grid method for the resolution of PDEs

AIChE Journal, 2003

An efficient interpolating wa®elet-based adapti®e-grid numerical method is described for sol®ing systems of bidimensional partial differential equations. The grid is dynamically adapted in both dimensions during the integration procedure so that only the rele-®ant information is stored, sa®ing allocation memory. The spatial deri®ati®es are directly calculated in a nonuniform grid using cubic splines. Numerical results for fi®e typical problems presented illustrate the efficiency and robustness of the method. The adapti®e strategy significantly reduces the computational times and the memory requirements, as compared to the fixed-grid approach.