A Finite Difference Scheme with Improved Boundary Approximation for the Heat Conduction Equation with Third-Type Boundary Conditions

Introduction. The development, analysis, and modification of finite difference schemes tailored to the specific features of the considered problem can significantly enhance the accuracy of modeling complex systems. In simulations of various processes, including hydrodynamic phenomena in shallow water bodies, it has been observed that for problems with thirdtype (Robin) boundary conditions, the theoretical error order of spatial discretization drops from second-order to firstorder accuracy, which in turn decreases the overall accuracy of the numerical solution. The present study addresses the relevant issue of how the approximation of third-type boundary conditions affects the accuracy of the numerical solution to the heat conduction problem. It also proposes a finite difference scheme with improved boundary approximation for the heat conduction equation with third-type boundary conditions and compares the accuracy of the numerical solutions obtained by the authors with known benchmark solutions.
Materials and Methods. The paper considers the one-dimensional heat conduction equation with third-type boundary conditions, for which an analytical solution is available. The problem is discretized, and it is shown that under standard boundary approximation, the theoretical order of approximation error for the second-order differential operator in the diffusion equation is O(h). To improve the accuracy of the numerical solution under specific third-type boundary conditions, a finite difference scheme is proposed. This scheme achieves second-order accuracy O(h2), for the differential operator not only at interior nodes but also at the boundary nodes of the computational domain.
Results. Test problems were used to compare the accuracy of numerical solutions obtained using the proposed scheme and those based on the standard boundary approximation.
Discussion and Conclusion. Numerical experiments demonstrate that the proposed scheme with enhanced boundary approximation for the heat conduction equation under specific third-type boundary conditions exhibits an effective accuracy order close to 2, which corresponds to the theoretical prediction. It is noteworthy that the scheme with standard boundary approximation also demonstrates an effective accuracy order close to 2, despite the lower theoretical order of boundary approximation. Importantly, the numerical error of the proposed scheme decreases significantly faster compared to the scheme with standard boundary treatment.

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