Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs
Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs
复制标题
DOI：
10.1016/j.camwa.2023.07.021
复制
发表时间：
2023-10
期刊：
影响因子：
--
通讯作者：
Giang D. Huynh;R. Abedi
中科院分区：
文献类型：
--
作者：
Giang D. Huynh;R. Abedi
我们给出了一维空间域和三个线性双曲、阻尼双曲型和抛物型偏微分方程组的时空DG方法。后两者对应于Maxwell-Cattneo-Vernotte(MCV)和傅立叶热传导问题。这种方法被称为帐篷-投手时空DG方法(TpSDG)，因为它与因果时空DG方法(CSDG)相似，在CSDG方法中，解通过投掷时空块来随时间推进。TpSDG方法将这种方法的适用范围从双曲型扩展到抛物型和双曲型偏微分方程组。对于具有空间均匀网格的问题，推导了一种传递矩阵方法，其中流入项、边界项和源项的值被映射到解系数和输出值。这类似于有限差分格式，但在空间元素的高斯点上有网格点，在时空中精度的顺序可以任意调节。该方法的谱稳定性分析为抛物线情形提供了稳定性修正因子。数值算例表明，该方法对具有非均匀材料性质的问题是适用的。
We present a spacetime DG method for 1D spatial domains and three linear hyperbolic, damped hyperbolic, and parabolic PDEs. The latter two correspond to Maxwell-Cattaneo-Vernotte (MCV) and Fourier heat conduction problems. The method is called the tent-pitcher spacetime DG method (tpSDG) due to its resemblance to the causal spacetime DG method (cSDG) wherein the solution advances in time by pitching spacetime patches. The tpSDG method extends the applicability of such methods from hyperbolic to parabolic and hyperbolic PDEs. For problems with a spatially uniform mesh, a transfer matrix approach is derived wherein the inflow, boundary, and source term values are mapped to the solution coefficient and output values. This resembles a finite difference scheme, but with grid points at the Gauss points of the spatial elements and arbitrarily tunable order of accuracy in spacetime. The spectral stability analysis of the method provides stability correction factors for the parabolic case. Numerical examples demonstrate the applicability of the method to problems with heterogeneous material properties.