We present a new approach to parallelization of the first-order backward difference discretization (BDF1) of the time derivative in partial differential equations, such as the nonlinear heat and viscous Burgers equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order 3-point central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at each Newton iteration is block bidiagonal, which can be inverted directly in a blockwise manner, thus leading to a set of fully decoupled equations associated with each time level. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations. The proposed parallel-in-time method preserves a quadratic rate of convergence of the Newton method of the sequential BDF1 scheme, so that the computational cost of solving each block matrix in parallel is nearly identical to that of the sequential counterpart at each time step. Numerical results show that the new parallel-in-time BDF1 scheme provides the speedup of up to 28 on 32 computing cores for the 2-D nonlinear partial differential equations with both smooth and discontinuous solutions.

翻译：本文提出了一种对偏微分方程（如非线性热方程和粘性Burgers方程）中时间导数项进行一阶后向差分离散化（BDF1）并行化的新方法。时间导数项采用基于隐式BDF1格式的直线法进行离散，而无粘项与粘性项则通过每个空间方向上一阶及二阶导数的传统二阶三点中心差分格式进行近似。时空域中的全局非线性离散方程组通过牛顿法对所有时间层进行同步求解。对于BDF1离散格式，每个牛顿迭代步的全时层系统呈现块双对角结构，可通过分块方式直接求逆，从而得到与各时间层完全解耦的方程组。这使得非线性微分方程的隐式BDF1离散化能够实现高效的并行时间计算。所提出的并行时间方法保持了顺序BDF1格式牛顿法的二次收敛速率，使得并行求解每个块矩阵的计算成本与顺序方法在每时间步的计算成本几乎相同。数值结果表明，对于具有光滑解与间断解的二维非线性偏微分方程，新型并行时间BDF1格式在32个计算核心上可实现最高28倍的加速比。