A controller executes a fuel introduction process when stopping combustion in a cylinder under a situation in which a crankshaft is rotating. The fuel introduction process causes a fuel injection valve to inject fuel and causes the fuel to flow out unburned from inside the cylinder to an exhaust passage. The opening degree of an exhaust gas recirculation valve at a point in time when a state in which the execution condition of the fuel introduction process is not satisfied is switched to a state in which the execution condition of the fuel introduction process is satisfied is a preliminary opening degree. During the execution of the fuel introduction process, the recirculation valve controlling section causes the opening degree of the exhaust gas recirculation valve to be smaller than the preliminary opening degree.