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<p>This paper presents a novel computational framework for solving the Rosenau–Burgers–biharmonic equation, a high-order nonlinear partial differential equation that models wave propagation in dispersive media with dissipative effects. The proposed approach combines Isogeometric Analysis (IGA) with high-order Explicit first stage, Singly Diagonally Implicit Runge–Kutta (ESDIRK) temporal discretization to address the significant challenges posed by the fourth-order biharmonic term. The IGA formulation leverages the inherent high-order continuity of Non-Uniform Rational B-Splines (NURBS) to achieve <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^2\)</EquationSource> </InlineEquation>-conformity without requiring mixed formulations or complex <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^1\)</EquationSource> </InlineEquation>-continuous finite elements. The adaptive refinement is realised through hierarchical B-splines (HB-splines), enabling local resolution of steep fronts while preserving the required high-order continuity. We establish the energy stability of the semi-discrete scheme and demonstrate optimal convergence rates in both space and time through rigorous numerical experiments. The method’s performance is validated through comparisons with traditional finite element approaches, adaptive refinement studies, and long-time dynamic simulations that confirm excellent agreement with asymptotic theoretical predictions. Our results show that the integrated IGA-ESDIRK framework offers superior accuracy, computational efficiency, and structure-preserving properties compared to existing numerical methods for this class of problems.</p>

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An energy-stable isogeometric analysis framework for the nonlinear Rosenau–Burgers–biharmonic equation

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Ujwal Warbhe

摘要

This paper presents a novel computational framework for solving the Rosenau–Burgers–biharmonic equation, a high-order nonlinear partial differential equation that models wave propagation in dispersive media with dissipative effects. The proposed approach combines Isogeometric Analysis (IGA) with high-order Explicit first stage, Singly Diagonally Implicit Runge–Kutta (ESDIRK) temporal discretization to address the significant challenges posed by the fourth-order biharmonic term. The IGA formulation leverages the inherent high-order continuity of Non-Uniform Rational B-Splines (NURBS) to achieve \(H^2\) -conformity without requiring mixed formulations or complex \(C^1\) -continuous finite elements. The adaptive refinement is realised through hierarchical B-splines (HB-splines), enabling local resolution of steep fronts while preserving the required high-order continuity. We establish the energy stability of the semi-discrete scheme and demonstrate optimal convergence rates in both space and time through rigorous numerical experiments. The method’s performance is validated through comparisons with traditional finite element approaches, adaptive refinement studies, and long-time dynamic simulations that confirm excellent agreement with asymptotic theoretical predictions. Our results show that the integrated IGA-ESDIRK framework offers superior accuracy, computational efficiency, and structure-preserving properties compared to existing numerical methods for this class of problems.