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Finite volume method for solving the stochastic Helmholtz equation
Advances in Difference Equations volume 2019, Article number: 84 (2019)
Abstract
In this paper, we consider the linear finite volume method (FVM) for the stochastic Helmholtz equation, driven by white noise perturbed forcing terms in onedimension. We first deduce the linear FVM for the deterministic Helmholtz problem. The dispersion equation is presented, and the error between the numerical wavenumber and the exact wavenumber is then analyzed. Comparisons between the linear FVM and the linear finite element method (FEM) are also made. The theoretical analysis and practical calculation indicate that the error of the linear FVM is half of that of the linear FEM. For the stochastic Helmholtz equation, convergence analysis and error estimates are given for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are provided to examine our theoretical results.
Introduction
In this paper, we consider the stochastic Helmholtz problem in onedimension driven by an additive white noise forcing term (see [6, 15, 18])
with the wavenumber k, where unknown u usually represents a pressure field in the frequency domain, \(i^{2}=1\), g is a deterministic real function with compact supports contained in \(I:=[0, 1]\), and \(\dot{W}(x)\) denotes the standard oneparameter family Brownian white noise that satisfies
where δ denote the usual Dirac δfunction and \(\mathbb {E}\) the expectation. Following the standard stochastic theory of the white noise [23, 27], we have
where \(\mathbb{V}\) is the variance operator. The stochastic Helmholtz equation has important applications in geophysics and medical science [4, 16, 18, 20, 26].
In recent years, finite difference methods (FDMs) and finite element methods (FEMs) have been developed to discretize the stochastic Helmholtz equation, for which the reader is referred to [5, 6, 8] and references cited therein. FDMs are easy to implement and locally conservative but not flexible to handle complex geometry. FEMs enjoy this flexibility. However, the main drawback of FEMs is its computational complexity and loss of the local conservation property. Finite volume methods (FVMs) possess the following advantages: the grid is flexible and the natural boundary conditions are easy to deal with; their implementation capability is comparable to that of finite difference methods; the mass conservation law is maintained, which is desirable in many engineering and science applications (see [10, 11, 13, 19] and references therein). It is worth mentioning that, for elliptic equations, although theoretical results indicate that both FVMs and FEMs enjoy the same optimal convergence order, the calculational effort of FVMs is usually less than that of FEMs, because of the asymmetry of their discrete schemes [19]. However, FVMs display many advantages when looking for the numerical solutions of computational fluid dynamics problems, because the mass conservation law is preserved [10]. The linear FVM for the stochastic Helmholtz equation in onedimension will be developed in this paper. Furthermore, theoretical results and practical computations will illustrate that the FVM is more efficient than the FEM when solving the deterministic Helmholtz equation. In particular, the error of the linear FVM is only half of that of the linear FEM.
To numerically solve the stochastic Helmholtz equation, we should consider two issues: one is randomness, and the other is a high wavenumber. We turn to randomness first. The stochastic Helmholtz equation is a stochastic partial differential equation. As presented in [1, 7, 12], the difficulty in the error analysis of general numerical methods for a stochastic partial differential equation is the lack of regularity of its solution. Particularly, for the onedimensional case, if Ẇ corresponds to the white noise, it has been shown that the regularity estimates are usually very weak, and lead to low order error estimates [1]. Therefore, we first follow [1, 7, 12] to approximate the white noise by a piecewise constant process, which converts the stochastic Helmholtz equation into the deterministic Helmholtz equation. For other methods to discretize the white noise, the reader is referred to [27] and references therein. We then apply the linear FVM to the stochastic Helmholtz equation with discretized white noise forcing terms. For the case with a more regular noise, highorder FVMs will be discussed in our future work.
On the other hand, solving the deterministic Helmholtz equation numerically with large wavenumbers is still a challenging task. For many years, FDMs (see [17, 22, 24, 25] and the reference therein), FEMs (see [2, 3, 15]) and discontinuous Galerkin methods (DGs) (see [14]) have been widely used to discrete the deterministic Helmholtz equation with various boundaries. For large wavenumbers, the quality of the numerical results usually deteriorates as the wavenumber k increases, which is the socalled “pollution effect” of high wavenumbers [3, 15]. Moreover, due to this pollution effect, the wavenumber of the numerical solution is different from that of the exact solution, which is known as “numerical dispersion” [15]. Hence, for numerically solving the deterministic Helmholtz equation, two main issues should be focused on: one is the numerical dispersion which is closely related to the pollution error, while the other is the solver cost. So far, the “pollution term” of the error estimates, which is connected with the “pollution effect” of high wavenumbers, and the numerical dispersion have not been considered for FVMs.
This paper is organized as follows. In Sect. 2, we deduce the linear FVM for the deterministic Helmholtz problem in onedimension, and then consider its solution’s existence and uniqueness. For the linear FVM, we establish its solution’s error estimates in \(H^{1}\) and \(L_{2}\)norm, then present its dispersion equation, and analyze the error between the numerical and exact wavenumbers. Comparisons between the linear FVM and FEM are also made in this section. Theoretical results indicate that the error for the linear FVM is half of that for the linear FEM. In Sect. 3, we study the approximation of (1.1) using discretized white noises. We also establish the regularity of the solution of the approximate problem and its error estimates in \(H^{1}\)norm. In Sect. 4, we study the linear FVM of the stochastic Helmholtz equation with discretized white noise forcing terms, and obtain the \(H^{1}\) error estimates between the finite volume solutions and the exact solution of (1.1). In Sect. 5, three numerical experiments, including two for the deterministic Helmholtz problem and one for the stochastic Helmholtz equation, are given to demonstrate the efficiency and accuracy of the linear FVM. In particular, numerical results illustrate that, when solving the deterministic Helmholtz problem, the error for the linear FVM is only half of that for the linear FEM. Finally, Sect. 6 contains the conclusions of this paper.
The linear FVM for the deterministic Helmholtz equation in onedimension
In this section, we deduce the linear FVM for the deterministic Helmholtz problem in onedimension
We choose the trial and the test function spaces as linear finite element and piecewise constant function spaces, respectively. For the linear FVM, we consider its solution’s existence and uniqueness, and establish the error estimates. The dispersion equation is also presented, and the error between the numerical and exact wavenumbers is analyzed. Furthermore, we make comparisons between the linear FVM and FEM. Theoretical analysis indicates that the error for the linear FVM is half of that for the linear FEM.
We begin with introducing some useful notations. We denote by \(L^{2}(I)\) the space of all squareintegrable complexvalued functions equipped with the inner product
and the norm
By \(H^{1}(I)\) we denote the Sobolev space
The norm of the space \(H^{1}(I)\) is defined as
We also introduce the \(H^{1}\)seminorm as
It follows from [15] that the Green’s function of (2.1) can be presented as
The solution \(u(x)\) of problem (2.1) exists for all \(k>0\) and can be written as
In addition, Lemma 1 in [15] presents bounds of the exact solution and its derivatives given the data g.
Lemma 2.1
Let \(u\in H^{2}(I)\) be the solution to problem (2.1) for given data \(g\in L^{2}(I)\). Then
Trial and test function spaces
In this subsection, we present the trial and the test function spaces for the linear FVM. We begin with discretizing the interval I into a grid \(T_{h}\) with nodes
Denote the length of the element \(I_{j}:=[x_{j1},x_{j}]\) by \(h_{j}=x_{j}x_{j1}\) and write \(h=\max_{1\leq j\leq n}h_{j}\). We assume the grid satisfies the quasiuniformity condition \(h_{j}\geq\mu h\) (\(j=1,2,\ldots,n\)) for some positive constant μ.
The trial space \(U_{h}\) is taken as the linear element space with respect to \(T_{h}\), which consists of all the functions \(u_{h}\) satisfying

(i)
\(u_{h}\in C(I)\), \(u_{h}(0)=0\) and

(ii)
\(u_{h}\) is linear on each \(I_{j}\) and is determined uniquely by its values at the endpoints of the element.
Obviously \(U_{h}\) is an ndimensional subspace of \(H^{1}_{E}(I) :=\{v:v\in H^{1}(I), v(0)=0\}\).
We next present the nodal basis functions for \(U_{h}\). The basis function with respect to \(x_{j}\) is
and
The functions \(\{\phi_{j}(x): j=1,2,\ldots,n\}\) form a basis of \(U_{h}\) and any \(u_{h}\in U_{h}\) has the following expression
where \(u_{j}=u_{h}(x_{j})\). On the element \(I_{j}\) we have
We then present a dual grid \(T_{h}^{*}\) with nodes
where \(x_{j1/2}= (x_{j1}+x_{j} )/2\), \(j=1,2,\ldots,n\). The dual elements are \(I_{0}^{*}=[x_{0},x_{1/2}]\), \(I_{j}^{*}=[x_{j1/2},x_{j+1/2}]\) (\(j=1,2,\ldots,n1\)), and \(I_{n}^{*}=[x_{n1/2},x_{n}]\). Accordingly we choose the test function space \(V_{h}\) as the piecewise constant function (step function) space, which contains all the functions \(v_{h}\in L^{2}(I)\) satisfying

(i)
\(v_{h}(x)=0\), for \(x\in I_{0}^{*}\), and

(ii)
\(v_{h}\) is a constant on each \(I_{j}^{*}\) (\(j=1,2,\ldots,n\)).
The basis functions of \(V_{h}\) are
Any \(v_{h}\in V_{h}\) has the form
where \(v_{j}=v_{h}(x_{j})\).
Variational formulation and linear FVM
We deduce the linear FVM for the deterministic problem (2.1) in this subsection. For this purpose, we first present the variational formulation of (2.1). A variational formulation of (2.1) can be obtained formally by multiplying the deterministic Helmholtz equation with \(v\in H_{E}^{1}(I)\). After partial integration we then arrive at the variational problem as follows: Find a function \(u \in H_{E}^{1}(I)\) such that
where
According to [15], the variational problem (2.8) has a unique weak solution. Applying Poincaré inequality, we obtain a continuity estimate for \(a(\cdot,\cdot)\), namely
We turn to the FVM for solving (2.1). The linear FVM approximation to (2.1) is: Find \(u_{h}=\sum_{m=1}^{n}u_{m}\phi_{m}(x)\) such that
where
Existence and uniqueness
In this subsection, we establish the existence and uniqueness of the solution for the linear FVM (2.9).
First, for \(u_{h}\in U_{h}\) it follows from (2.6) that
Next, we define an interpolation operator \(\varPi^{*}_{h} :H_{E}^{1}(I)\rightarrow V_{h}\) by
We then have the lemma as follows.
Lemma 2.2
As the homogeneous equation
admits only the trivial solution, there exists a constant \(\alpha>0\) independent of the subspace \(U_{h}\) such that for sufficiently small kh,
Proof
We first rewrite \(a(u_{h},\varPi^{*}_{h}\omega_{h})\) as follows:
where
Below we examine the positive definiteness of \(a_{1}(u_{h},\varPi^{*}_{h}u_{h})\). By (2.5),
Thus, \(k^{2}(u_{h},\varPi^{*}_{h}u_{h})\) is a real number. Furthermore, by (2.9) we have
Therefore,
By the interpolation theory, we get
Then we combine (2.14) with (2.15) to obtain
Hence, for sufficiently small kh, one has
Now we turn to showing (2.13). Suppose by contradiction that there exists a sequence \(\{\tilde{u}_{h}\}, \tilde{u}_{h}\in U_{h}\), satisfying
Since \(H_{E}^{1}(I)\) is weakly sequentially compact, \(\{\tilde{u}_{h}\}\) has a subsequence (again written as \(\{\tilde{u}_{h}\}\)) which converges weakly to some \(\tilde{u}\in H_{E}^{1}(I)\). Take any \(\omega\in H_{E}^{1}(I)\) and write \(\varPi_{h}\omega\) as the interpolation projection of ω onto the trial function space \(U_{h}\). It is clear that \(\varPi^{*}_{h} (\omega\varPi_{h}\omega )=0\). It follows from the interpolation theory that, when h is sufficiently small,
By (2.17) we have
On the other hand, we next prove that \(a(\tilde{u}_{h},\varPi^{*}_{h} \omega\omega)\rightarrow0\), \(h\rightarrow0\). We first rewrite \(a(\tilde{u}_{h},\varPi^{*}_{h} \omega\omega)\) as follows:
It follows from the continuity of \(a(u,v)\) and the interpolation theory that
In addition,
It follows from the Cauchy inequality and the interpolation theory that
Combining (2.20) and (2.21) with (2.24) yields
We combine (2.19) with (2.25) to obtain
For fixed \(\omega\in H_{E}^{1}(I)\), \(a(u,\omega)\) is a bounded linear functional on \(H_{E}^{1}(I)\), which implies
The assumption of the lemma then implies \(\tilde{u}=0\). So the sequence \(\tilde{u}_{h}\) converges weakly to zero. From the compactness of the imbedding of \(H_{E}^{1}(I)\) in \(L^{2}(I)\), we know that \(\tilde{u}_{h}\) converges strongly to zero in \(L^{2}(I)\), which gives
Furthermore, it follows from the Cauchy inequality and the interpolation theory that
Finally, by (2.17) and (2.29), we conclude
This contradicts (2.16) and completes the proof. □
Based on Lemma 2.2, the following theorem indicates that the solution of (2.9) exists and is unique.
Theorem 2.3
If h is sufficiently small, then the linear FVM (2.9) has a unique solution for any given \(g\in L^{2}(I)\).
Proof
By virtue of the wellknown results in linear algebra, we only need to show that the homogeneous equation
admits only the trivial solution, which follows from (2.13). □
Convergence order estimates
In this subsection, we present estimates for the error \(uu_{h}\) in \(H^{1}\) and \(L_{2}\)norm for the linear FVM (2.9). The following theorem establishes an estimate for the error \(uu_{h}\) in \(H^{1}\)norm.
Theorem 2.4
Let u be solution of (2.1) satisfying \(u\in H^{2}(I)\) and \(u_{h}\) be the solution of the linear FVM (2.9). If h is sufficiently small, then
Proof
Clearly, we have
Together with Lemma 2.2 and the interpolation theory, we observe that
Let \(\omega_{j}=\omega_{h}(x_{j})\). Then we have \(\varPi^{*}_{h} \omega_{h}=\sum_{j=1}^{n}\omega_{j}\psi_{j}\) and
By the Cauchy and Hölder inequalities and the definition of \(\varPi_{h}\), we have that
Below we present estimates for \(a(u\varPi_{h}u,\varPi_{h}^{*}\omega_{h})\) based on the above inequality, which leads to an estimate for \(u_{h}\varPi_{h}u_{1}\). It follows from (2.6) that
Furthermore, by the mean value theorem, there exists \(\xi_{0}\in I_{j}\) such that
Hence, we deduce that
which yields
Therefore, we have
In addition, by the interpolation theory we have
Combining (2.33) with (2.37)–(2.39) yields
Applying the interpolation theory in Sobolev spaces leads to
The above two estimates and the regularity of u (see Lemma 2.1) lead to (2.31) and completes the proof. □
The above theorem indicates that the solution \(u_{h}\) of the linear FVM (2.9) approximates the solution u of (2.1) to first order in \(H^{1}\)norm. Moreover, the term associated with \(k^{2}h^{2}\) presents the pollution effect, which depends on the wavenumber k. The next theorem establishes an estimate for the error \(uu_{h}\) in \(L_{2}\)norm.
Theorem 2.5
Let \(u_{h}\) be the solution of (2.9), and u be the solution of (2.1) with \(u\in H_{E}^{1}(I)\cap W^{3,1}(I)\). If \(k^{2}h\) is small enough, then
Proof
Let us introduce an auxiliary problem: For a given \(e=uu_{h}\), find \(\omega\in H_{E}^{1}(I)\) such that
It follows from Lemma 2.1 that the above problem possesses a unique solution satisfying
Combining (2.41) with (2.32) leads to
By (2.31),(2.42) and the continuity of \(a(\cdot,\cdot)\), we have
Moreover, by a simple computation, we get
Thus there holds
Applying the Taylor expansion with an integral remainder yields
In addition, we have
Combining (2.47)–(2.49) yields
By (2.43), (2.44), (2.50) and noting the imbedding relation \(W^{3,1}(I)\rightarrow H^{2}(I)\), for sufficiently small \(k^{2}h\), we have
This validates the estimate (2.40) and completes the proof. □
The above theorem indicates that in \(L_{2}\)norm the solution \(u_{h}\) of the linear FVM (2.9) approximates the solution u of (2.1) to second order. Moreover, the term associated with \(kh^{2}\) presents the pollution effect, which depends on the wavenumber k.
Error analysis between the numerical and exact wavenumbers
In this subsection, we obtain the dispersion equation for the linear FVM (2.9) by a classical dispersion analysis, and provide an error analysis between the numerical and exact wavenumbers. Comparisons between the linear FVM and FEM are also made in this subsection, which indicate that the error for the linear FVM is half of that for the linear FEM.
Assuming that a uniform mesh is used, we then rewrite the linear FVM (2.9) for “interior” points \(x_{j}=j/n\), \(1< j< n\) as
where
Following the classical harmonic approach, we next insert the discrete expression of a plane wave \(u_{j} := e^{ikx_{j}}\) into equation (2.51). By a simple computation, we get the dispersion equation
Replacing variable k in the parameters \(A_{s}\), \(A_{o}\) with the numerical wavenumber \(k^{N}\) in equation (2.52) yields an equation for the exact wavenumber k and the numerical wavenumber \(k^{N}\), namely
Based on the above equation, we will analyze the error between the numerical wavenumber \(k^{N}\) and the exact wavenumber k in the following proposition, when kh is small enough.
Proposition 2.6
For the linear FVM (2.9), if kh is small enough, then
Proof
Let \(\tau:= kh\), and denote
Applying Taylor expansions for \(f_{1}(\tau)\) and \(\frac{1}{f_{2}(\tau)}\) at the point \(\tau= 0\) yields
In addition, from equation (2.53), we have
Together with equations (2.55) and (2.56), we have
Based on the above equation, applying the Taylor expansion of the function \(\sqrt{1+\tau}\) at the point \(\tau=0\) leads to the conclusion of this proposition. □
The above proposition indicates that \(k^{N}\) approximates k to second order. Moreover, the term associated with \(k^{3}h^{2}\) presents the pollution effect, which depends on the wavenumber k. For the linear finite element method, a similar estimate for the relation \(k^{N}\) and k is obtained; see Remark 2.7.
Remark 2.7
For the linear finite element method, if kh is sufficiently small, then
Equations (2.54) and (2.57) indicate that \(k^{N}\) approximates k to second order, for both the linear FVM and FEM. Moreover, the term associated with \(k^{3}h^{2}\) presents the pollution effect. We also find that the term associated with \(k^{3}h^{2}\) for the linear FVM is half of that for the linear FEM, when kh is small enough. However, when a uniform mesh is used, the quadratic finite volume method may not behave as well as the quadratic finite element method; see Remark 2.8.
Remark 2.8
Assume that a uniform mesh is used. For the quadratic finite volume method, when kh is sufficiently small, we have
For the quadratic finite element method, when kh is small enough, we get
We next present the normalized numerical phase and group velocities for the linear FVM, which are two important tools for measuring the numerical dispersion (see [17, 21]). In practice, the former is usually preferred. For a numerical method, when its normalized numerical phase velocity approximates 1 better, its numerical dispersion is smaller, and its accuracy is higher. Similar conclusions hold for the normalized numerical group velocity. For the convenience of analysis, let v be the velocity of propagation, ω be the angular frequency, λ be the wavelength, and G be the number of gridpoints per wavelength, that is, \(G=\frac{\lambda}{h}\). Since \(\lambda=\frac{2\pi v}{\omega}\) and \(k=\frac{\omega}{v}\), we have \(kh=\frac{2\pi}{G}\). Together with equation (2.53), we conclude that
The normalized numerical phase velocity, which is equivalent to \(\frac {k^{N}}{k}\) (cf. [9, 17]), is
In addition, together with (2.54), we get
which indicates that the normalized numerical phase velocity approximates 1 to second order, when \(kh\rightarrow0\), and \(\frac{1}{48}k^{2}h^{2}\) is the main error term. Furthermore, the normalized numerical group velocity for the linear FVM is
Figure 1 shows the normalized phase and group velocity curves for the linear FVM and FEM, respectively. It is easy to find that the curves for the linear FVM approximate 1 better than those for the linear FEM. Specifically, the error between the normalized phase velocity for the linear FVM and 1 is almost half of that for the linear FEM. The reason may be that, for the normalized phase velocity, the coefficient of the main error term for the linear FVM is half of that for the linear FEM (see (2.62)). Therefore, we expect that the linear FVM will enjoy higher numerical accuracy, when compared with the linear FEM. This will be illustrated by two numerical experiments in the next section.
The approximation problem for the stochastic Helmholtz equation in onedimension
In this section, we first introduce an approximate problem of (1.1) by replacing the white noise Ẇ by its piecewise constant approximation \(\dot{W}^{s}\). Then we establish the regularity of the solution of the approximate problem and its error estimates.
We discretize the interval I in the same way as it is done in Sect. 2.1. Let
for each interval \(I_{j}\). It is wellknown that \(\{\xi_{I_{j}}\}\) is a family of independent identically distributed normal random variables with mean 0 and variance 1 (see [23]). Then the piecewise constant approximation to \(\dot{W}(x)\) is given by
where \(\chi_{I_{j}}\) is the characteristic function of \(I_{j}\). It is easy to see that \(\dot{W}^{s}(x)\in L^{2}(I)\). However, the following lemma shows that the \(L_{2}\)norm \(\\dot{W}^{s}\_{0}\) of \(\dot{W}^{s}\) is unbounded as \(h\rightarrow0\).
Lemma 3.1
There holds
Proof
It is easy to see that
Therefore, we have that
By using the quasiuniformity condition \(h_{j}\geq\mu h\) (\(j=1,2,\ldots ,n\)), we come to the conclusion of this lemma. □
Replacing \(\dot{W}(x)\) by \(\dot{W}^{s}(x)\) in (1.1), we have the following stochastic Helmholtz equation with a discretized white noise forcing term:
The variational problem of (3.3) is as follows: Find a function \(u^{s} \in H_{E}^{1}(I)\) such that
As \(\dot{W}^{s}\in L^{2}(I)\), it follows from [15] that (3.4) has a unique solution \(u^{s}\). We then establish an estimate for the error \(uu^{s}\), where u is the solution of (1.1).
Lemma 3.2
There is a unique solution \(u^{s}\in H^{2}(I)\) of problem (3.3) which satisfies
where \(C_{5}\) is a positive constant independent of h.
Proof
It follows from Lemma 2.1 that
By (3.6) and (3.2), we come to the desired result. □
Next we estimate the error between the weak solution u of (1.1) and its approximation \(u^{s}\). To this end, we present the solution u of (1.1) and the solution \(u^{s}\) of (3.3) by the Green function (2.2) as
We then establish the regularity of the Green function \(G(x,s)\) and \(\frac{\partial G(x,s)}{\partial x}\) defined in (2.2) in the following lemma, which will play an important role in the error estimate between u and \(u^{s}\).
Lemma 3.3
There hold
Proof
Assume that \(0\leq y< z\leq1\). We first prove that (3.9) holds. Obviously, we have
By (2.2), we get that
Similarly, we obtain
Combining (3.11) and (3.13) with (3.12) yields (3.9).
Below we prove (3.10). By (2.2), we have that
Arguing as before, we have
It follows from (3.14) that
Combining the above three inequalities, we obtain the desired estimate (3.10). □
Now we establish an error estimate between u and \(u^{s}\).
Theorem 3.4
Let u and \(u^{s}\) be the solutions of (1.1) and (3.3), respectively. We have
Proof
By combining (3.7) with (3.8), we observe that
Applying Itô’s isometry yields
It follows from the Hölder inequality that
Then the desired result (3.18) follows from (3.10) and (3.20). □
Finite volume method for the stochastic Helmholtz equation in onedimension
In this section, we consider the finite volume approximation of variational problem (3.4) and establish its error estimates.
The linear finite volume approximation to (3.4) is: Find \(u_{h}^{s}=\sum_{m=1}^{n}u_{m}^{s}\phi_{m}(x)\) such that
The approximate variational problem (4.1) has a unique solution. The following theorem presents an error estimate for \(uu_{h}^{s}\).
Theorem 4.1
Let u and \(u_{h}^{s}\) be the solutions of (1.1) and (4.1), respectively. If kh is small enough, then we have
Proof
By (2.31), (3.5), (3.18) and the triangle inequality, we have that
which leads to the desired result. □
Numerical experiments
In this section, we present numerical examples to demonstrate our theoretical results in the previous section. All the experiments are performed with Matlab 7v on an Intel Xeon (4core) with 3.60 GHz and 16 GB RAM.
Problem 1
Consider the deterministic Helmholtz equation
The exact solution of the above problem is
We use this problem to measure the accuracy for two schemes, including the linear FVM and FEM. The error is measured in the relative discrete \(L_{2}\)norm and the relative discrete \(H^{1}\)seminorm. In details, the discrete \(L_{2}\)norm and \(H^{1}\)seminorm are respectively defined as: for any complex vector \(\mathbf{z}= [z_{0},z_{1},\ldots ,z_{M} ]\),
where \(z_{j}\) is the complex modulus of \(z_{j}\). Let u and \(u_{h}\) denote the exact and numerical solutions, respectively. Then, the relative discrete \(L_{2}\)norm is defined as \(\frac{\uu_{h}\_{0}}{\u\ _{0}}\). The relative discrete \(H^{1}\)seminorm is defined in a similar way.
Tables 1, 2 and 3 show the error in the relative discrete \(L_{2}\)norm for two different schemes with different gridpoints N for \(k=30,200,500\), respectively. We find that, for both the linear FVM and FEM, the rate of convergence for the error in the relative discrete \(L_{2}\)norm is of order 2, as expected. In addition, Tables 4, 5 and 6 show the error in the relative discrete \(H^{1}\)seminorm for two different schemes with different gridpoints N for \(k=30,200,500\), respectively. It is easy to see that, for both schemes, the rate of convergence for the error in the relative discrete \(H^{1}\)seminorm is of order 2. In particular, from these six tables, we see that the error for the linear FVM is only half of that for the linear FEM, which verifies the relations between \(k^{N}\) and k for the two methods as presented previously in (2.54) and (2.57).
Further comparison between the linear FEM and FVM is given in Tables 7 and 8. Table 7 presents the error in the relative discrete \(L_{2}\)norm of two schemes for the case \(kh=0.125\), and Table 8 shows the corresponding error in the relative discrete \(H^{1}\)seminorm. The wavenumber k in the two tables varies form 200 to 700. As seen from these two tables, the accuracy of the linear FVM is higher than that of the linear FEM, and it deteriorates much slower if kh is chosen to be a constant.
Problem 2
We solve the deterministic Helmholtz problem
The above problem’s exact solution is:
By this problem, the accuracy of the linear FVM and FEM are also measured in the relative discrete \(L_{2}\)norm and the relative discrete \(H^{1}\)seminorm.
Tables 9 and 10 show the error in the relative discrete \(L_{2}\)norm for two different schemes with different gridpoints N for \(k=30,300\), respectively. In addition, Tables 11 and 12 show the error in the relative discrete \(H^{1}\)seminorm for two different schemes with different gridpoints N for \(k=30,300\), respectively. From these four tables, we know that for both the linear FVM and FEM, the convergence rate of the error in the relative discrete \(L_{2}\)norm or the relative discrete \(H^{1}\)seminorm is 2. Furthermore, seen from Tables 9–12, we find that the error for the linear FVM is only half of that for the linear FEM, which is an interesting result. This confirms the efficiency of the linear FVM.
Problem 3
We consider the stochastic Helmholtz equation
When white noise is absent, the above problem reduces to Problem 1. We will use the random number generator to simulate the Gaussian random process \(\dot{W}^{s}\). Furthermore, we shall follow [8] to evaluate \(E(u_{h}^{s})\) by using the Monte Carlo method when examining
to ensure that we have used enough samples. Notice that it is impossible to evaluate \(E (uu^{s}_{h}_{1}^{2} )\), since it is impossible obtain an explicit expression for u. We also employ the following type of error:
to check the error estimates for the finite volume method. By a simple computation, we have
When \(k=1\), \(E (u_{1}^{2} )=0.8456\). In addition, \(E (u_{1}^{2} )=0.5157\) for \(k=6\), and \(E (u_{1}^{2} )=0.5047\) for \(k=12\).
The computational results of the linear FVM approximations for (5.3) with \(k=1,6,12\) are displayed in Tables 13, 14 and 15, respectively. The third columns of the tables show that the convergence rate for \(E(u_{h}^{s})\) is of order 1, which confirms our theoretical result presented in the previous section. The sixth columns of the tables show that the rate of convergence for \(E(u_{h}^{s}_{1}^{2})\) is of order 1 as expected, which implies that our sample sizes are good enough to ensure the accuracy of the Monte Carlo method.
Conclusions
In this paper, we proposed the linear FVM for the stochastic Helmholtz equation, driven by an additive white noise forcing term in onedimension. Firstly, the linear FVM for the deterministic Helmholtz equation in onedimension was presented, and then its solution’s existence and uniqueness were considered. For the linear FVM, its solution’s error estimate in \(H^{1}\) and \(L_{2}\)norm were established. Moreover, its dispersion equation was presented, and the error between the numerical and exact wavenumbers was analyzed. We also made comparisons between the linear FVM and FEM. Theoretical analysis and practical computations indicated that the error for the linear FVM is only half of that for the linear FEM. By means of approximating the white noise by a piecewise constant process, we converted the stochastic Helmholtz equation into the deterministic Helmholtz equation, which is an approximate problem for the stochastic Helmholtz problem. The regularity of the solution for the approximate problem was discussed, and its error estimates in \(H^{1}\)norm were presented. Furthermore, the linear FVM was applied for solving this approximate problem, and the \(H^{1}\) error estimates between the finite volume solutions and the exact solution of the stochastic Helmholtz problem were obtained. Finally, numerical experiments were given to verify our theoretical results.
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Acknowledgements
The authors would like to thank Prof. Zhongying Chen in Sun YatSen University for valuable comments, which led to a much better version of this article.
Funding
This research is partially supported by a project of Shandong Province Higher Educational Science and Technology Program of China under the grant J18KA221, the Natural Science Foundation of China under grants 11301310, 11471196 and 11771257.
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Xu, R., Wu, T. Finite volume method for solving the stochastic Helmholtz equation. Adv Differ Equ 2019, 84 (2019). https://doi.org/10.1186/s136620192011x
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Keywords
 Stochastic Helmholtz equation
 White noise
 Finite volume method