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.
翻译：现代计算科学和工程问题的多重复杂性可以阻止在多维模拟中使用传统的数字方法。 因此,在这些情况下,需要采用新的算法来解决局部差异方程式(PDEs),这些方程式的特点在广泛的空间和时间尺度上不断变化。 为了应对这些挑战,我们提出了一个多分辨率波列算法,用大量的数据压缩和明确的错误控制来解决PDEs。我们通过投射字段和空间衍生物操作器将空间分解到波盘基功能上。我们提供了对字段及其衍生物的波盘表示的错误估计。然后,我们用我们的估算法来构建一个稀疏多分辨率分解,保证了规定的准确性。此外,我们在时间整合中嵌入一个预测-校正程序,以动态地调整计算网格,并随着PDE的演变而保持其解决方案的准确性。我们举例来突出我们方法的准确性和适应性。
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