## 1 Introduction  
Diesel engines are widely utilized in transportation due to their high thermal efficiency, strong torque output, and superior fuel economy. However, their exhaust emissions present significant environmental and health challenges. Compared to gasoline engines, diesel engines release higher levels of nitrogen oxides (NOxs) and particulate matter (PM), which contribute to air pollution and pose health risks. In response, governments worldwide have implemented stricter regulations on vehicle emissions, emphasizing the need for advancements in PM reduction technologies [1, 2, 3, 4]. A key solution for mitigating PM emissions is the diesel particulate filter (DPF). Typically constructed from porous ceramic materials such as silicon carbide (SiC) or cordierite, the DPF features a honeycomb-like structure designed to capture and filter PM from exhaust gases [5; 6]. As illustrated in Figure 1, the filter consists of alternating sealed and open channels: exhaust gases enter through unsealed pathways, where PM is captured on the filter walls, while the cleaned exhaust exits through the outlet. Although DPF walls generally have a thickness of 300-400 \(\upmu\)m, PM deposition predominantly occurs within the first few tens of micrometers from the surface. As PM accumulates over time, the pressure drop across the filter increases, imposing a greater load on the engine. To maintain efficiency, PM must be periodically removed through an oxidation process known as regeneration (Figure 2). Carbon-based PM typically combusts at approximately 650 \({}^{\circ}\)C; however, given that diesel exhaust temperatures range between 100 and 500 \({}^{\circ}\)C, additional fuel is often required to elevate exhaust temperatures to enable PM oxidation [7; 8; 9]. when PM oxidation catalysts are supported on a DPF. The figures below illustrate the differences in catalytic activity depending on the form in which the catalyst is applied. They compare particulate catalysts (Figure 3a) with catalysts supported on a DPF (Figure 3b), highlighting how these configurations affect the oxidation process of PM. In the case of particulate catalysts, catalyst particles are attached directly to PM, creating multiple points of contact between the catalyst and PM. This direct interaction enhances the oxidation reaction, leading to a higher frequency factor for catalytic activity. The oxidation reaction occurs efficiently because the catalyst is near the carbonaceous PM. In the case of the catalyst supported on the DPF, the catalyst is coated on the surface of the DPF rather than being directly attached to PM. When PM accumulates on the DPF wall, only a portion of it comes into direct contact with the catalyst. As a result, oxidation is more dependent on gas-phase reactions, and the catalytic activity is generally lower compared to the particulate catalyst case. This configuration leads to a lower frequency factor, as not all PM interacts directly with the catalyst. These differences play a crucial role in optimizing DPF design and improving the efficiency of PM oxidation processes in diesel after-treatment systems. In addition, various experiments and simulations related to DPF have been conducted in the past. However, these were limited to experiments and numerical calculations on parts of a DPF. Furthermore, it is not easy to experiment with the performance of various catalysts on DPFs under various conditions. This study aims to analyze the behavior of PM deposition and oxidation processes in DPFs with various catalysts and to identify the optimal range for effective PM oxidation catalysts. In this study, we have numerically calculated the performance of the PM oxidation of various catalysts supported on a DPF. To achieve our goal, we made three improvements to the previous model, subdividing the soot oxidation process in more detail. First, we assumed the existence of two oxidation reactions. The first is the reaction between the catalyst and the soot (solid-state reaction: SSR).