We consider partitioned time integration for heterogeneous coupled heat equations. First and second order multirate, as well as time-adaptive Dirichlet-Neumann waveform relaxation (DNWR) methods are derived. In 1D and for implicit Euler time integration, we analytically determine optimal relaxation parameters for the fully discrete scheme. Similarly to a previously presented Neumann-Neumann waveform relaxation (NNWR) first and second order multirate methods are obtained. We test the robustness of the relaxation parameters on the second order multirate method in 2D. DNWR is shown to be very robust and consistently yielding fast convergence rates, whereas NNWR is slower or even diverges. The waveform approach naturally allows for different timesteps in the subproblems. In a performance comparison for DNWR, the time-adaptive method dominates the multirate method due to automatically finding suitable stepsize ratios. Overall, we obtain a fast, robust, multirate and time adaptive partitioned solver for unsteady conjugate heat transfer.

中文翻译：

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异构耦合热方程的时间自适应多速率Dirichlet-Neumann波形弛豫方法

我们考虑异构耦合热方程的分区时间积分。推导出一阶和二阶多速率以及时间自适应 Dirichlet-Neumann 波形弛豫 (DNWR) 方法。在 1D 和隐式欧拉时间积分中，我们通过分析确定完全离散方案的最佳松弛参数。类似于先前提出的诺依曼-诺依曼波形弛豫 (NNWR) 一阶和二阶多速率方法。我们在二维中测试二阶多速率方法上松弛参数的稳健性。DNWR 表现出非常稳健且始终如一地产生快速收敛速度，而 NNWR 较慢甚至发散。波形方法自然允许子问题中的不同时间步长。在 DNWR 的性能比较中，由于自动寻找合适的步长比，时间自适应方法优于多速率方法。总的来说，我们获得了一个快速、稳健、多速率和时间自适应的非定常共轭传热分区求解器。