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<p>We present a second-order numerical method for a class of nonlocal systems of conservation laws in multiple space dimensions. The proposed scheme employs a second-order accurate spatial reconstruction coupled with an appropriate time discretization. The presence of nonlocal terms and the complexity of accurately incorporating them into the numerical scheme make the problem intricate, both analytically and computationally, which we address through a carefully designed discretization strategy. As an essential contribution of this work, we analytically prove that the resulting scheme is positivity-preserving, a crucial property for ensuring the physical validity of quantities such as density, which must remain nonnegative. In addition, we prove that the scheme is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{L}^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>L</mtext> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>- stable. Numerical experiments are conducted for two problems: a crowd dynamics model and the Keyfitz–Kranzer system. The results illustrate the superior performance of the second-order method compared to that of a first-order implementation and confirm the theoretical analysis.</p>

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A positivity-preserving second-order scheme for multi-dimensional systems of nonlocal conservation laws

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Nikhil Manoj,
  • G. D. Veerappa Gowda,
  • K. Sudarshan Kumar

摘要

We present a second-order numerical method for a class of nonlocal systems of conservation laws in multiple space dimensions. The proposed scheme employs a second-order accurate spatial reconstruction coupled with an appropriate time discretization. The presence of nonlocal terms and the complexity of accurately incorporating them into the numerical scheme make the problem intricate, both analytically and computationally, which we address through a carefully designed discretization strategy. As an essential contribution of this work, we analytically prove that the resulting scheme is positivity-preserving, a crucial property for ensuring the physical validity of quantities such as density, which must remain nonnegative. In addition, we prove that the scheme is \(\textrm{L}^\infty \) L ∞ - stable. Numerical experiments are conducted for two problems: a crowd dynamics model and the Keyfitz–Kranzer system. The results illustrate the superior performance of the second-order method compared to that of a first-order implementation and confirm the theoretical analysis.