The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. To meet these challenges, we present a multiresolution wavelet algorithm to solve PDEs with significant data compression and explicit error control. We discretize in space by projecting fields and spatial derivative operators onto wavelet basis functions. We provide error estimates for the wavelet representation of fields and their derivatives. Then, our estimates are used to construct a sparse multiresolution discretization which guarantees the prescribed accuracy. Additionally, we embed a predictor-corrector procedure within the temporal integration to dynamically adapt the computational grid and maintain the accuracy of the solution of the PDE as it evolves. We present examples to highlight the accuracy and adaptivity of our approach.

中文翻译：

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非线性偏微分方程的一种多分辨率自适应小波方法

现代计算科学与工程问题的多尺度复杂性使得传统数值方法无法用于多维模拟。因此，在这些情况下需要新的算法来解决偏微分方程 (PDE)，其特征在广泛的空间和时间尺度上演化。为了应对这些挑战，我们提出了一种多分辨率小波算法来解决具有显着数据压缩和显式误差控制的 PDE。我们通过将场和空间导数算子投影到小波基函数上来实现空间离散。我们为场及其导数的小波表示提供误差估计。然后，我们的估计用于构建稀疏多分辨率离散化，以保证规定的准确性。此外，我们在时间积分中嵌入了一个预测器-校正器程序，以动态调整计算网格，并随着 PDE 的发展保持解的准确性。我们提供了一些例子来强调我们方法的准确性和适应性。