The Local Radial Basis Function Collocation Method with Automatic Differentiation for Nonlinear Problems

doi:10.1016/j.camwa.2026.10.020

The Local Radial Basis Function Collocation Method with Automatic Differentiation for Nonlinear Problems

2026-10-23

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DOI：10.1016/j.camwa.2026.10.020

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This paper presents an automatic differentiation–enhanced local radial basis function collocation method (AD-LRBFCM) for solving strongly nonlinear partial differential equations (PDEs). Within a meshfree framework, local radial basis functions are employed to construct sparse approximations without the need for numerical integration, providing a direct collocation-based discretization. By incorporating automatic differentiation (AD), the residuals and Jacobian matrices can be evaluated automatically from the implemented discrete equations, avoiding the tedious and error-prone process of manual differentiation. The resulting nonlinear system is solved using Newton’s iteration, with the AD-generated Jacobian used in the linearized system at each iteration. The effectiveness of the proposed method is validated through benchmark problems, including nonlinear heat conduction and large-deformation hyperelastic solid mechanics. Numerical results show excellent agreement with reference solutions, demonstrating good accuracy and stability for the problems considered. Overall, the AD-LRBFCM provides a flexible computational framework for solving complex nonlinear PDEs.

The Local Radial Basis Function Collocation Method with Automatic Differentiation for Nonlinear Problems

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