A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of numerical methods ADER-DG using a local DG predictor for solving ODE and PDE systems. The basis functions were chosen in the form of Lagrange interpolation polynomials with nodal points at the roots of the right Radau polynomials, which differs from the classical formulations of the ADER-DG method, where it is customary to use the roots of Legendre polynomials. It was shown that the use of this basis leads to A-stability and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2769_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-stability in the case of using the DAE solver as ODE solver. The numerical method ADER-DG allows one to obtain a highly accurate numerical solution even on very coarse grids, with a step greater than the main characteristic scale of solution variation. The local discrete time solution can be used as a numerical solution of the DAE system between grid nodes, thereby providing subgrid resolution even in the case of very coarse grids. The classical test examples, including simple DAE systems of indices 1, 2, 3, were solved by developed numerical method ADER-DG. It was found that in the case of DAE systems of index 1, the empirical convergence orders p approximately correspond to the expected convergence orders that occur in the case of solving ODE systems. With increasing index of the DAE system, a decrease in the empirical convergence orders p is observed. Using the method of decreasing the index of the DAE system led to an increase in the empirical convergence orders—the situation when the empirical convergence orders decrease with increasing index of the DAE system is expected. An unexpected result was obtained in the numerical solution of the stiff DAE system—the empirical convergence orders of the numerical solution obtained using the developed method turned out to be significantly higher than the values expected for this method in the case of stiff problems. It turns out that the use of Lagrange interpolation polynomials with nodal points at the roots of the right Radau polynomials is much better suited for solving stiff problems. This result is of fundamental importance for the choice of basis functions when using the ADER-DG method with a local DG predictor to solve stiff problems. Using the method of decreasing the index of the DAE system led to an increase in the empirical convergence orders. Estimates of the computational costs showed that the computational costs of the ADER-DG numerical method with a local DG predictor are approximately comparable to the computational costs of implicit Runge–Kutta methods used to solve DAE systems. Methods were proposed to reduce the computational costs of the ADER-DG numerical method.

High Order ADER-DG Method with Local DG Predictor for Solutions of Differential-Algebraic Systems of Equations

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发表日期：

2025年1月4日

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Ivan S. Popov

摘要

A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of numerical methods ADER-DG using a local DG predictor for solving ODE and PDE systems. The basis functions were chosen in the form of Lagrange interpolation polynomials with nodal points at the roots of the right Radau polynomials, which differs from the classical formulations of the ADER-DG method, where it is customary to use the roots of Legendre polynomials. It was shown that the use of this basis leads to A-stability and $L_{1}$ $L_{1}$ -stability in the case of using the DAE solver as ODE solver. The numerical method ADER-DG allows one to obtain a highly accurate numerical solution even on very coarse grids, with a step greater than the main characteristic scale of solution variation. The local discrete time solution can be used as a numerical solution of the DAE system between grid nodes, thereby providing subgrid resolution even in the case of very coarse grids. The classical test examples, including simple DAE systems of indices 1, 2, 3, were solved by developed numerical method ADER-DG. It was found that in the case of DAE systems of index 1, the empirical convergence orders p approximately correspond to the expected convergence orders that occur in the case of solving ODE systems. With increasing index of the DAE system, a decrease in the empirical convergence orders p is observed. Using the method of decreasing the index of the DAE system led to an increase in the empirical convergence orders—the situation when the empirical convergence orders decrease with increasing index of the DAE system is expected. An unexpected result was obtained in the numerical solution of the stiff DAE system—the empirical convergence orders of the numerical solution obtained using the developed method turned out to be significantly higher than the values expected for this method in the case of stiff problems. It turns out that the use of Lagrange interpolation polynomials with nodal points at the roots of the right Radau polynomials is much better suited for solving stiff problems. This result is of fundamental importance for the choice of basis functions when using the ADER-DG method with a local DG predictor to solve stiff problems. Using the method of decreasing the index of the DAE system led to an increase in the empirical convergence orders. Estimates of the computational costs showed that the computational costs of the ADER-DG numerical method with a local DG predictor are approximately comparable to the computational costs of implicit Runge–Kutta methods used to solve DAE systems. Methods were proposed to reduce the computational costs of the ADER-DG numerical method.