def multistep_deis_second_order_update(
        self,
        model_output_list: List[torch.Tensor],
        *args,
        sample: torch.Tensor = None,
        **kwargs,
    ) -> torch.Tensor:
        """
        One step for the second-order multistep DEIS.

        Args:
            model_output_list (`List[torch.Tensor]`):
                The direct outputs from learned diffusion model at current and latter timesteps.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.Tensor`:
                The sample tensor at the previous timestep.
        """
        timestep_list = args[0] if len(args) > 0 else kwargs.pop("timestep_list", None)
        prev_timestep = args[1] if len(args) > 1 else kwargs.pop("prev_timestep", None)
        if sample is None:
            if len(args) > 2:
                sample = args[2]
            else:
                raise ValueError("missing `sample` as a required keyword argument")
        if timestep_list is not None:
            deprecate(
                "timestep_list",
                "1.0.0",
                "Passing `timestep_list` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        if prev_timestep is not None:
            deprecate(
                "prev_timestep",
                "1.0.0",
                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma_t, sigma_s0, sigma_s1 = (
            self.sigmas[self.step_index + 1],
            self.sigmas[self.step_index],
            self.sigmas[self.step_index - 1],
        )

        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
        alpha_s1, sigma_s1 = self._sigma_to_alpha_sigma_t(sigma_s1)

        m0, m1 = model_output_list[-1], model_output_list[-2]

        rho_t, rho_s0, rho_s1 = sigma_t / alpha_t, sigma_s0 / alpha_s0, sigma_s1 / alpha_s1

        if self.config.algorithm_type == "deis":

            def ind_fn(t, b, c):
                # Integrate[(log(t) - log(c)) / (log(b) - log(c)), {t}]
                return t * (-np.log(c) + np.log(t) - 1) / (np.log(b) - np.log(c))

            coef1 = ind_fn(rho_t, rho_s0, rho_s1) - ind_fn(rho_s0, rho_s0, rho_s1)
            coef2 = ind_fn(rho_t, rho_s1, rho_s0) - ind_fn(rho_s0, rho_s1, rho_s0)

            x_t = alpha_t * (sample / alpha_s0 + coef1 * m0 + coef2 * m1)
            return x_t
        else:
            raise NotImplementedError("only support log-rho multistep deis now")

        One step for the second-order multistep DEIS.

        Args:
            model_output_list (`List[torch.Tensor]`):
                The direct outputs from learned diffusion model at current and latter timesteps.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.Tensor`:
                The sample tensor at the previous timestep.
        
src/diffusers/schedulers/scheduling_deis_multistep.py