Scholarlink

<p>The weighted compact central schemes (WCCS) (Shen <i>et al.</i>, J. Comput. Phys., 466:111370, 2022) are capable of achieving arbitrarily uniform spatio-temporal high-order accuracy on a compact stencil with a single explicit time step, and of capturing shock waves without Riemann solvers. However, WCCS need to calculating the space-time derivatives via the Cauchy-Kovalevskaya (CK) process which requires the information of the Jacobian matrix, the Hessian tensor, and even higher-order derivative tensors of the fluxes depending on the order of the scheme. When the hyperbolic system contains several nonlinear equations, high-order derivative tensors could become extremely cubersome. To address this issue, we modify the Jacobian-free CK (JFCK) process for Lax-Wendroff WENO schemes (Zorío <i>et al.</i>, <i>J. Sci. Comput.</i>, 71(2): 246–273, 2017) by utilizing the compactness of WCC schemes and construct Jacobian-free WCCS (WCCS-JFCK). Unlike the JFCK process for WENO finite difference schemes, which uses flux values at several mesh points to compute flux derivatives, the modified JFCK process for WCCS employs Taylor expansions to derive the flux values at virtual points from the information in the local cell, and then uses finite difference to compute flux derivatives, thereby retaining the compactness of WCCS. Compared to the original WCCS using exact CK process (WCCS-CK), WCCS-JFCK are more straightforward to implement, and are readily generalized to general complex hyperbolic systems. Theoretical analysis and numerical experiments demonstrate that the WCCS-JFCK can ensure high-order accuracy in both space and time, while maintaining the same numerical stability as WCCS-CK.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Jacobian-free high-order weighted compact central schemes for hyperbolic conservation laws

  • 期刊论文

  • 发表日期：

    2025年9月25日

查看全文
  • Run Li,
  • Hua Shen

摘要

The weighted compact central schemes (WCCS) (Shen et al., J. Comput. Phys., 466:111370, 2022) are capable of achieving arbitrarily uniform spatio-temporal high-order accuracy on a compact stencil with a single explicit time step, and of capturing shock waves without Riemann solvers. However, WCCS need to calculating the space-time derivatives via the Cauchy-Kovalevskaya (CK) process which requires the information of the Jacobian matrix, the Hessian tensor, and even higher-order derivative tensors of the fluxes depending on the order of the scheme. When the hyperbolic system contains several nonlinear equations, high-order derivative tensors could become extremely cubersome. To address this issue, we modify the Jacobian-free CK (JFCK) process for Lax-Wendroff WENO schemes (Zorío et al., J. Sci. Comput., 71(2): 246–273, 2017) by utilizing the compactness of WCC schemes and construct Jacobian-free WCCS (WCCS-JFCK). Unlike the JFCK process for WENO finite difference schemes, which uses flux values at several mesh points to compute flux derivatives, the modified JFCK process for WCCS employs Taylor expansions to derive the flux values at virtual points from the information in the local cell, and then uses finite difference to compute flux derivatives, thereby retaining the compactness of WCCS. Compared to the original WCCS using exact CK process (WCCS-CK), WCCS-JFCK are more straightforward to implement, and are readily generalized to general complex hyperbolic systems. Theoretical analysis and numerical experiments demonstrate that the WCCS-JFCK can ensure high-order accuracy in both space and time, while maintaining the same numerical stability as WCCS-CK.