On a Method for the Numerical Integration of Differential Equations
On a Method for the Numerical Integration of Differential Equations
Sahakyan Robert,
Yenokyan Robert
Summary
Summary
Key words: difference equations, computational grid, forward sweep, backward sweep, correctness, system of equations
This paper examines a boundary value problem for a linear differential equation where the boundary conditions are insufficient to determine the solution y(x) starting from one end of the interval. Typically, the right boundary condition is replaced by an auxiliary condition at the left end, and a solution is found that satisfies the equation and the left boundary condition but not the right one; the solution is then adjusted. However, this conventional approach often leads to a significant loss of accuracy.
The proposed study introduces the forward and backward sweep method, which avoids the accumulation of computational errors. The essence of the method lies in propagating the boundary condition, specified at the left end, throughout the computational interval, thus ensuring the stability and correctness of the solution. The correctness of the method is proven for the case of constant coefficients, and it is shown that similar properties hold for variable coefficients as well.
The proposed algorithm is easy to implement computationally and provides highly accurate solutions with minimal computational cost. The method can be applied to a wide range of linear and quasi-linear boundary value problems, including problems of heat conduction, continuum mechanics, and mathematical physics.
DOI: https://doi.org/10.58726/27382923-2025.2-95