# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
###### Abstract  
The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore,  
improvement method to increase the time accuracy and improve the stability of the FD algorithms [5; 6; 7; 8; 9; 10]. Additionally, stencil-based methods for increasing the temporal accuracy usually use smaller computational amounts than Lax-Wendroff methods with a similar calculation accuracy [11; 12; 13]; however, the calculation of the coefficients of these stencils is more complicated. These methods are mainly used for acoustic field simulation; however, there are a few literature reports on boundary value problems such as heat conduction problems due to the difficulty of boundary treatment. At the same time, some researchers have studied ADER methods, which are another class of Lax-Wendroff methods [14; 15; 16] to solve the computational flow dynamics (CFD) problem. The implicit (compact) HOFD algorithms often used in thermal simulation significantly improve the simulation stability and computational accuracy [17; 18; 19]. The explicit FD algorithms typically use less computational cost than the implicit FD ones. Therefore, we prefer to study the explicit HOFD algorithm to improve the numerical simulation stability and accuracy and reduce computational costs. In addition, a small number of scholars have explored releasing the upper limit of the time step restrained by the stability condition of the explicit FD algorithm based on the spatial filtering method [20; 21; 22] and the eigenvalue-type method [23; 24; 25], however, these need a large amount of memory.  
As mentioned above, the Lax-Wendroff HOFD algorithms have been studied in many fields and have achieved good results. The implicit HOFD algorithms are mostly used in thermal simulation, while the Lax-Wendroff HOFD algorithm has been poorly studied.  
In order to improve the accuracy and stability of the existing HOFD algorithm, first, this paper presents a Lax-Wendroff high-order finite difference (LHOFD) algorithm to numerically solve the 2D transient heat conduction equations (HCEs). Using the stability analysis and operator combination optimization strategy, an improved LHOFD (IHOFD) algorithm was developed to improve the stability of the LHOFD algorithm. Secondly, we develop the formulas of general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as general high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative. A unified formula for general HOCFD and HONFD coefficients that can calculate the spatial derivative of any order is also derived. In the domain of wavenumbers, we analyze the accuracy of the HOCFD operator of the fourth-order spatial derivative and derive the stability formulas of the algorithms. Then, we use a strategy of combining the high-order non-central and central FD approximation with the same high-order accuracy as the internal computational domain to calculate the derivatives of the boundary regions. Finally, the stability, accuracy, and effectiveness of the proposed algorithms are analyzed by examples.