We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in both the spatial and temporal dimensions simultaneously. We also propose a novel wavelet-based recursive algorithm to reduce the system sensitivity stemming from steep initial and/or boundary conditions. The resulting nonlinear equations are solved using the Newton-Raphson method. We parallelize the construction of the tangent operator along with the solution of the system of algebraic equations. We perform rigorous verification studies using the nonlinear Burgers' equation. The application of the method is demonstrated solving Sod shock tube problem using the Navier-Stokes equations. The numerical results of the method reveal high-order convergence rates for the function as well as its spatial and temporal derivatives. We solve problems with steep gradients in both the spatial and temporal directions with a priori error estimates.

翻译：我们提出了一种高阶时空小波方法，用于以用户指定的精度求解非线性偏微分方程。该技术利用小波理论和先验误差估计，在空间和时间维度上同时离散化问题。我们还提出了一种新颖的基于小波的递归算法，以降低由陡峭初始和/或边界条件引起的系统敏感性。所得非线性方程组采用牛顿-拉弗森方法求解。我们并行构建切线算子并求解代数方程组。我们使用非线性Burgers方程进行了严格的验证研究。通过求解基于Navier-Stokes方程的Sod激波管问题，展示了该方法的应用。该方法的数值结果表明，函数及其空间和时间导数均具有高阶收敛速度。我们求解了在空间和时间方向上均具有陡峭梯度的问题，并提供了先验误差估计。