Four terms in the Taylor expansion (Equation ( 23)) are taken, and a general component is written out for the IIF2 scheme (Equation ( 15)). In order to derive the leading-order terms of both temporal and spatial truncation errors, we localize the IIF schemes. ![]() For a global scheme, a numerical value u i n + 1 at the time level t n + 1 depends on all numerical values at the last time level t n. IIF schemes are exponential integrator-type schemes and they are global. Specifically, the second-order Krylov IIF scheme has the following form: From our numerical experiments in this paper, we can see that our numerical schemes have already given a clear accuracy order with very small sizes M ≪ N, and M does not need to be increased when the spatial–temporal resolution is refined. ![]() The value of M is taken to be large enough such that the errors of Krylov subspace approximations are much less than the truncation errors of the numerical schemes (Equation ( 12)). We note that V M, 0, n, V M, 1 − r, n and V M, i, n, i = 2 − r, 3 − r, ⋯, − 1 are orthonormal bases of different Krylov subspaces for the same matrix C, which are generated with different initial vectors in the Arnoldi algorithm. Where γ 0, n = ∥ U n + Δ t n ( α n R → ( U → n ) + β n F → ( U → n ) ) ∥ 2, V M, 0, n and H M, 0, n are the orthonormal basis and upper Hessenberg matrix generated by the Arnoldi algorithm with the initial vector U n + Δ t n ( α n R → ( U → n ) + β n F → ( U → n ) ) γ 1 − r, n = ∥ F → ( U → n + 1 − r ) ∥ 2, V M, 1 − r, n and H M, 1 − r, n are orthonormal basis and upper Hessenberg matrix generated by the Arnoldi algorithm with the initial vector F → ( U → n + 1 − r ) γ i, n = ∥ α n + i R → ( U → n + i ) + β n + i F → ( U → n + i ) ∥ 2, V M, i, n and H M, i, n are the orthonormal basis and upper Hessenberg matrix generated by the Arnoldi algorithm with the initial vectors α n + i R → ( U → n + i ) + β n + i F → ( U → n + i ), for i = 2 − r, 3 − r, ⋯, − 1. The large sparse matrix C is projected to the Krylov subspace: Here we briefly describe how to use the Krylov subspace approximation to compute the product of a matrix exponential and a vector (e.g., e C Δ t v →). This approach has been used in the Krylov IIF schemes for solving reaction–diffusion systems and convection–diffusion equations. In order to efficiently implement the IIF schemes for fourth-order PDEs (Equations ( 12), ( 15) and ( 16)), we use the Krylov subspace method to approximate the product of a matrix exponential and a vector. ![]() Hence at every time-step, we need to find the products of matrix exponentials and vectors. Additionally, time-step sizes in the IIF schemes for fourth-order PDEs (Equations ( 12), ( 15) and ( 16)) can be non-uniform in general. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.įor solving PDEs defined on two- or higher-spatial-dimension domains, directly computing and storing exponential matrices in IIF schemes is very expensive and impractical for a typical computer. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. We analyze the truncation errors of the fully discretized schemes. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. ![]() In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. The methods can be designed for an arbitrary order of accuracy. In, IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature.
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