Explicit solution of high-dimensional parabolic PDEs: Application of Kronecker product and vectorization operator in the Haar wavelet method

In this paper, we propose a numerically stable and efficient method based on Haar wavelets for solving high-dimensional second-order parabolic partial differential equations (PDEs). In the proposed framework, the spatial second-order derivatives in the governing equation are approximated using the Haar wavelet series. These approximations are subsequently integrated to obtain the corresponding lower-order derivatives. By substituting these expressions into the governing equation, the PDE is transformed into a system of first-order ordinary differential equations. This resulting system is then advanced in time using Euler's scheme.

Conventional Haar wavelet methods transform the given PDEs into a system with a large number of equations, which makes them computationally expensive. In contrast, the present Haar wavelets method (HWM) significantly reduces the number of algebraic equations. Moreover, the incorporation of the Kronecker product and vectorization operator properties in the HWM substantially decreases the computational cost compared to existing Haar wavelet methods in the literature (e.g., [25], [34], [35]). The HWM achieves second-order accuracy in spatial variables. We demonstrate the effectiveness of the HWM through various multi-dimensional problems, including two-, three-, four-, and ten-dimensional cases. The numerical results confirm the accuracy and efficiency of the proposed approach.