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<p>This paper investigates the ultra-weak discontinuous Galerkin method for solving two-dimensional diffusive-viscous wave equations with variable coefficients, which are crucial for modeling wave propagation in fluid-saturated porous media. By carefully selecting numerical fluxes, the UWDG method is proven to be energy stable in the sense that the discrete energy is dissipative. The optimal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm error estimate is obtained by adopting the ultra-weak projection technique and a novel supercloseness result for the projection errors. Moreover, several fully discrete schemes which adopt implicit-explicit backward difference formulas for time integration are designed and their unconditional energy stability and optimal error estimates are investigated. Numerical experiments confirm the accuracy and reliability of the proposed methods for both smooth and discontinuous media.</p>

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Ultra-Weak Discontinuous Galerkin Methods for Two-Dimensional Diffusive-Viscous Wave Equations

  • 期刊论文

  • 发表日期：

    2026年1月31日

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  • Haijin Wang,
  • Qiang Zhang,
  • Yuan Xu

摘要

This paper investigates the ultra-weak discontinuous Galerkin method for solving two-dimensional diffusive-viscous wave equations with variable coefficients, which are crucial for modeling wave propagation in fluid-saturated porous media. By carefully selecting numerical fluxes, the UWDG method is proven to be energy stable in the sense that the discrete energy is dissipative. The optimal \(L^2\) L 2 -norm error estimate is obtained by adopting the ultra-weak projection technique and a novel supercloseness result for the projection errors. Moreover, several fully discrete schemes which adopt implicit-explicit backward difference formulas for time integration are designed and their unconditional energy stability and optimal error estimates are investigated. Numerical experiments confirm the accuracy and reliability of the proposed methods for both smooth and discontinuous media.