A high order, block finite difference, error inhibiting scheme for the transport equation

TL;DR

The paper develops error-inhibiting block finite difference schemes for the 1D transport equation to mitigate long-time error growth caused by phase errors. It constructs a high-order method that attains fourth-order accuracy for short to moderate times and six-order convergence for long times, with the strategy rooted in manipulating truncation-error dynamics; it also shows the scheme is equivalent to a nodal-based DG method with $p=1$. Through eigenvalue/eigenvector analysis, asymptotic expansions, and numerical experiments across parameter choices, the authors demonstrate stability and long-time accuracy improvements, including effective post-processing to enhance short-time accuracy. This work links BFD schemes with DG formulations, offering a pathway to high-order, long-time accurate transport solvers and suggesting extensions to non-periodic boundaries and higher dimensions, thereby enhancing practical long-time simulations in hyperbolic problems.

Abstract

We propose a block finite difference, error inhibiting scheme that is fourth-order accurate for short to moderate times and has a six-order convergence rate for long times. This scheme outperforms the standard fourth-order Finite Difference scheme. We also demonstrate that the proposed scheme is a particular type of nodal-based Discontinuous Galerkin method with $p=1$.

Figures (5)

• Figure 1: Scheme \ref{['eq2']} Convergence plots, $\log_{10} \| {{\bf E}} \|$ vs. $\log_{10} h$ ,for different values of $c_1,c_2$. Final time, $T=1$ -a: no post-processing; b: spectral post-processing
• Figure 2: Scheme \ref{['eq2']} Convergence plots, $\log_{10} \| {{\bf E}} \|$ vs. $\log_{10} h$ ,for different values of $c_1,c_2$. Final time, $T=1.1$ -a: no post-processing; b: spectral post-processing
• Figure 3: Scheme \ref{['eq2']}, Convergence plots, $\log_{10} \| {{\bf E}} \|$ vs. $\log_{10} h$ - Periodic BC for $c_{1}=c_{2}$, with post processing, for different final times. a: $T=100$, b: $T=1000$.
• Figure 4: The exact numerical solutions for $u(x,t)=\sin(4\pi(x-t))$, final time $T=4800$ and $N=32$ with post processing. a: standard fourth-order scheme, $c_{1}=c_{2}=0$, b: $c_{1}=c_{2}=1/2$.
• Figure 5: The evolution of the error in time for the standard second, fourth, and six-order finite difference schemes and the BFD scheme, \ref{['eq2']} , with $c_1=c_2=1/2$ with and without post-processing. a: for $N=128$ (256 degrees of freedom), b: for $N=16$ (32 degrees of freedom).