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Numerical resolution of the wave equation using the spectral method

Abstract

We present a new procedure for the numerical study of the wave equation. We use the spectral discretization method associated with the Euler scheme for spatial and temporal discretization. A detailed numerical analysis leads to an a priori error estimate. We confirm the high precision of the method presented by a numerical study.

Introduction

During the last decade, many works have focused on the numerical analysis of second-order parabolic and hyperbolic partial differential equations such as the heat and wave equations [18].

The time and space approximation for the wave equation has been studied in [9] using the finite element method. We present a new study of this problem using the spectral method associated with backward Euler discretization scheme. The spectral method is known for its high precision [10, 11]. For the space discretization, the discrete spaces are constructed from spaces of polynomials of high degree. Then the discrete problem is obtained using the Galerkin method combined with numerical integration.

The outline of the paper is as follows. In Sect. 2, we study the continuous problem and present some energy estimate properties. In Sect. 3, we are interested to the time-semidiscrete problem. We discretize the second time derivative by using a second difference quotient of the solution on a nonuniform temporal grid. Then we obtain an optimal a priori error estimate. In Sect. 4, we study the fully discrete problem and establish an optimal a priori error estimate. Finally, in Sect. 5, we present a numerical study.

The continuous problem

Consider an open bounded connected domain \(\Omega \subset {\mathbb{R}}^{d}\) (\(d=2\) or 3) with Lipschitz continuous boundary Γ, and let T be a positive real number. Let \(H^{s}(\Omega )\), \(s>0\), be the Sobolev spaces associated with the norm \(\| \cdot \| _{s,\Omega }\) and seminorm \(| \cdot | _{s,\Omega }\). The space \(H^{1}_{0}(\Omega )\) is the closure in the space \(H^{1}(\Omega )\) of infinitely differentiable functions with compact support in Ω, and \(H^{-1}(\Omega )\) is its dual space. We denote by \((\cdot ,\cdot )\) and \(\| \cdot \| \), respectively, the scalar product and associated norm in \(L^{2}(\Omega )\). By \(H^{\frac{1}{2}}(\partial \Omega )\) we denote the trace space of functions in \(H^{1}(\Omega )\). Let \(\gamma \subset \partial \Omega \), \(H^{\frac{1}{2}}_{00}(\gamma )\) be the space of functions in \(H^{\frac{1}{2}}(\gamma )\) such that their extension by zero to \(\partial \Omega /\gamma \) belongs to \(H^{\frac{1}{2}}(\partial \Omega )\).

In the following, we define

  • \(u({\mathbf{x}},t)\) on \(\Omega \times\, ]0,T[\) as

    $$ \begin{aligned} u:{}&]0,T[ \,\longrightarrow X \\ &t\longmapsto u(t)=u(\cdot ,t), \end{aligned} $$

    where X is a separable Banach space.

  • \(\mathcalligra{C}^{j}(0,T;X)\) represents the set of functions of time of class \(\mathcalligra{C}^{j}\) with values in X. It is a Banach space with norm

    $$ \Vert u \Vert _{\mathcalligra{C}^{j}(0,T;X)}=\sup_{0\leq t\leq T}\sum _{l=0}^{j} \bigl\Vert \partial _{t}^{l} u \bigr\Vert _{X}, $$

    where \(\partial _{t}^{l} u\) is the time partial derivative of u of order l.

  • \(L^{p}(0,T;X)=\{v \text{ mesurable on } ]0, T[ \text{ such that } \int _{0}^{T}\|v(t)\|^{p}_{X} \,dt<\infty \}\) is the Banach space with norm

    $$ \Vert v \Vert _{L^{p}(0,T;X)}= \textstyle\begin{cases} (\int _{0}^{T} \Vert v(t) \Vert ^{p}_{X} \,dt)^{\frac{1}{p}} & \text{for } 1\leq p < + \infty , \\ \sup_{0\leq t\leq T} \Vert v(t) \Vert _{X} & \text{for } p = + \infty . \end{cases} $$
  • \(H^{s}(0,T;X)=\{v \in L^{2}(0,T;X); \partial ^{k} v\in L^{2}(0,T;X); k\leq s\}\) is the Hilbert space with scalar product

    $$ (u,v)=\Biggl((u,v)_{L^{2}(0,T;X)} + \sum_{k=0}^{s} \bigl(\partial ^{k} u, \partial ^{k} v \bigr)_{L^{2}(0,T;X)}\Biggr)^{\frac{1}{2}}. $$
  • \(W^{m,1}(0,T,X)\) is the space of functions in \(L^{1}(0,T,X)\) such that all their derivatives up to the order m belong to \(L^{1}(0,T,X)\).

Consider the wave equation problem

$$ \textstyle\begin{cases} {\partial _{t}^{2} u} -\Delta u =0& \text{in } \Omega \times\, ]0,T[, \\ u =0 & \text{on } \Gamma \times\, ]0,T[, \\ u(\cdot,0)=u_{0} & \text{in } \Omega , \\ {\partial _{t} u}(\cdot ,0)=v_{0} & \text{in } \Omega , \end{cases} $$
(1)

where the wave u is the unknown defined on \(\Omega \times\, ]0,T[\), and \((u_{0},v_{0})\) are the data functions defined on Ω.

This problem can be written in a more general form,

{ t Φ ( 0 1 Δ 0 ) Φ = F in  Ω × ] 0 , T [ , u = 0 on  Γ × ] 0 , T [ , Φ ( , 0 ) = Φ 0 in  Ω ,
(2)

where Φ=( u v = t u ), F=( f g ), and Φ 0 =( u 0 v 0 ).

Lemma 1

If \((f,g)\in L^{1}(0,T;H^{1}_{0}(\Omega ))\times L^{1}(0,T;L^{2}(\Omega ))\) and \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\), then

$$ \bigl( \Vert v \Vert ^{2} + \Vert \nabla u \Vert ^{2} \bigr)^{\frac{1}{2}}\leq \bigl( \Vert v_{0} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} \bigr)^{\frac{1}{2}} + \int _{0}^{t} \bigl( \Vert f \Vert + \Vert g \Vert \bigr) (s)\,ds, \quad 0\leq t \leq T. $$
(3)

Proof 1

Taking the inner product of the first equation of problem (2) and ( Δ u v ) and integrating by parts the second term, we obtain

$$ {\frac{1}{2}} {\frac{d }{dt}}\bigl( \Vert \partial _{t}u \Vert ^{2}\bigr) + {\frac{1}{2}} { \frac{d }{dt}}\bigl( \Vert \nabla u \Vert ^{2}\bigr)\leq \Vert f \Vert + \Vert g \Vert . $$

By integrating this inequality between 0 and t we get estimate (3).

Remark 1

Consider the following Laplace problem:

$$ \textstyle\begin{cases} -\Delta u=h & \text{in } \Omega , \\ u =0 & \text{on } \Gamma . \end{cases} $$
(4)

Let \(u=(\Delta )^{-1}h\) be the solution of problem (4). The operator \((\Delta )^{-1}\) is a self-adjoint positive definite isometry of the space \(H^{-1}(\Omega )\) into \(H^{1}_{0}(\Omega )\). Thus, for \(h\in H^{-1}(\Omega )\), we obtain \(\|{((\Delta )^{-1})}^{\frac{1}{2}}h\|=\|h\|_{H^{-1}(\Omega )}\) (see [12], Chap. 1, Thm. 12.3, for the proof). Then, for \((f,g) \in L^{1}(0,T,L^{2}(\Omega ))\times L^{1}(0,T,H^{-1}(\Omega ))\), taking the inner product of the first equation of system (2) and ( u ( Δ ) 1 v ), we obtain the following estimate:

$$ \bigl( \Vert v \Vert _{H^{-1}(\Omega )}^{2} + \Vert u \Vert ^{2} \bigr)^{\frac{1}{2}}\leq \bigl( \Vert v_{0} \Vert _{H^{-1}( \Omega )}^{2} + \Vert u_{0} \Vert ^{2} \bigr)^{\frac{1}{2}} + \int _{0}^{t} \bigl( \Vert f \Vert + \Vert g \Vert _{H^{-1}( \Omega )}\bigr) (s)\,ds. $$

Finally, we have the following result proved in [12, Chap. 1].

Proposition 1

For any data \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\), system (1) has a unique solution \(u \in \mathcalligra{C}^{1}(0,T;L^{2}(\Omega ))\cap \mathcalligra{C}^{0}(0,T;H^{1}_{0}( \Omega ))\). Moreover, this solution satisfies

$$ \Vert \partial _{t} u \Vert ^{2} + \Vert \nabla u \Vert ^{2} = \Vert \nabla u_{0} \Vert ^{2} + \Vert u_{0} \Vert ^{2} . $$

Discretization on time

Consider a partition of the time interval \([0,T]\) into subintervals \([t_{k},t_{k+1}]\), \(1\le k \le I\), such that \(0=t_{0}< t_{1} < \cdots <t_{K}=T\). We denote \(\delta t_{k}=t_{k+1}-t_{k}\), \(\delta t=(\delta t_{1},\ldots ,\delta t_{K})\), and \(|\delta t|= \max_{1\le k \le K}|\delta t_{k}|\).

To formulate the time semidiscrete problem, we apply the Euler implicit method to system (1). Then it consists in finding the sequence of functions \({(u^{k})}_{0\leq k\leq K}\) in the space \(H^{1}_{0}(\Omega )\times L^{2}(\Omega )\times {H^{1}_{0}(\Omega )}^{K-1}\) such that

$$ \textstyle\begin{cases} \frac{u^{k+1}-u^{k}}{\delta t_{k}}- \frac{u^{k}-u^{k-1}}{\delta t_{k-1}} -\delta t_{k}\Delta u^{k+1}=0 & \text{in } \Omega ,\ 1\leq k\leq K, \\ u^{k+1} =0 & \text{on } \Gamma ,\ 1\leq k\leq K, \\ u^{0}=u_{0} & \text{in } \Omega , \\ u^{1}=u_{0}+\delta t_{0}v_{0} & \text{in } \Omega . \end{cases} $$
(5)

We suppose that the data \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\). Then, if \(u^{0}\) and \(v^{0}\) are known, then we easily show that \(u^{k+1}\), \(k\geq 1\), is a solution of the following weak problem:

Find \(u^{k+1}\in H^{1}_{0}(\Omega )\) such that for all \(v \in H^{1}_{0}(\Omega )\),

$$ \begin{aligned} &\int _{\Omega } u^{k+1}({\mathbf{x}}) v({\mathbf{x}}) \,d{\mathbf{x}}+ \delta t_{k}^{2} \int _{\Omega }\nabla u^{k+1}({\mathbf{x}}) \nabla v({ \mathbf{x}})\,d{\mathbf{x}} \\ &\quad = \int _{\Omega } \biggl(u^{k} + \frac{\delta t_{k}}{\delta t_{k-1}} \bigl(u^{k}-u^{k-1}\bigr) \biggr) ({\mathbf{x}})v({ \mathbf{x}}) \,d{\mathbf{x}}. \end{aligned} $$
(6)

Proposition 2

If \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\) is known, then problem (6) has a unique solution \(u^{k+1}\), \(k\geq 1\), in \(H^{1}_{0}(\Omega )\). Moreover, the solution \({(u^{k})}_{0\leq k\leq K}\) of problem (5) verifies for \(0\leq k\leq K\) the following stability condition:

$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq \Vert v_{0} \Vert ^{2} +2 \Vert \nabla u_{0} \Vert ^{2} +2\delta t_{0}^{2} \Vert \nabla v_{0} \Vert ^{2}. $$
(7)

Proof 2

Using the Lax–Milgram theorem, we show that problem (6) has a unique solution. Then, by iteration on k, we deduce that problem (5) has a unique solution.

Taking the inner product of \(\frac{u^{k+1}-u^{k}}{\delta t_{k}}\) and the first equation in (5), we obtain

$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2}=\biggl( \frac{u^{k+1}-u^{k}}{\delta t_{k}}, \frac{u^{k}-u^{k-1}}{\delta t_{k-1}}\biggr)+\bigl(\nabla u^{k+1},\nabla u^{k}\bigr). $$
(8)

Applying the Cauchy–Schwarz inequality leads to

$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq \biggl\Vert \frac{u^{k}-u^{k-1}}{\delta t_{k-1}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}. $$
(9)

Then by iteration on k we have

$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq \biggl\Vert \frac{u^{1}-u^{0}}{\delta t_{0}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{1} \bigr\Vert ^{2}. $$

Finally, we conclude by using the third and fourth equations of system (5).

Remark 2

1) We notice that for \(k\geq 1\), the solution \(u^{k+1}\) of problem (6) belongs to \(H^{s+1}(\Omega )\), \(s\geq {\frac{1}{2}}\). When the domain Ω is convex or of dimension 1, \(s\geq 1\) is explicitly known. For \({\frac{1}{2}}\leq s \leq 1\), from condition (7) we derive the inequality

$$ \bigl\Vert u^{k+1} \bigr\Vert ^{2} \leq C \delta t_{k}^{-2s} \bigl( \Vert v_{0} \Vert ^{2} +2 \Vert \nabla u_{0} \Vert ^{2} +2\delta t_{0}^{2} \Vert \nabla v_{0} \Vert ^{2} \bigr), $$
(10)

where the constant C is independent of the step δt.

This inequality is not optimal since \(\| u^{k+1}\| ^{2}\) is not bounded independently of the step δt.

2) Using the implicit Euler scheme for the time discretization, problem (2) is written as follows: Find Φ k =( u k v k ) such that,

{ Φ k + 1 Φ k δ t k ( 0 1 Δ 0 ) Φ k + 1 = F k + 1 in  Ω , 0 k K , u k + 1 = 0 on  Γ , 0 k K , Φ 0 = Φ 0 in  Ω ,
(11)

where F k + 1 =( f k + 1 g k + 1 ).

For \(n\neq 0\), systems (11) and (5) coincide if \(F^{k+1}={\mathbf{0}}\), \(k\geq 1\). When \(n=0\), we propose the following two cases where the two systems completely coincide:

  1. i)

    We replace the fourth equation of system (5) by the following implicit equation:

    $$ \textstyle\begin{cases} u^{1}- \delta t_{0}^{2}\Delta u^{1}=u_{0} + \delta t_{0}v_{0} & \text{in } \Omega , \\ u^{1}=0 & \text{on } \Gamma. \end{cases} $$
    (12)
  2. ii)

    We replace the third equation of system (11):

    { Φ k + 1 Φ k δ t k ( 0 1 Δ 0 ) Φ k + 1 = F k + 1 in  Ω , 1 k K , u k + 1 = 0 on  Γ , 1 k K , v 1 = ( u 0 + δ t 0 v 0 v 0 ) in  Ω .
    (13)

    Multiplying the first equation in (13) by ( Δ u k + 1 u k + 1 ), we obtain the following stability condition:

    $$ \bigl\Vert v^{k+1} \bigr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq 2 \bigl( \bigl\Vert v^{1} \bigr\Vert ^{2} + \bigl\Vert \nabla u^{1} \bigr\Vert ^{2} \bigr) +2 \Biggl(\sum _{j=1}^{k} \delta t_{j} \bigl( \bigl\Vert g^{j+1} \bigr\Vert + \bigl\Vert \nabla f^{j+1} \bigr\Vert \bigr) \Biggr)^{2}. $$
    (14)

    However, if we take the inner product of the first equation of system (13) and ( u k + 1 ( Δ ) 1 u k + 1 ), we obtain the following stability condition in terms of weaker norms:

    $$ \begin{aligned} &\bigl\Vert v^{k+1} \bigr\Vert _{H^{-1}(\Omega )}^{2} + \bigl\Vert u^{k+1} \bigr\Vert ^{2} \\ &\quad \leq 2 \bigl( \bigl\Vert v^{1} \bigr\Vert _{H^{-1}(\Omega )}^{2} + \bigl\Vert u^{1} \bigr\Vert ^{2} \bigr) +2 \Biggl( \sum_{j=1}^{k} \delta t_{j} \bigl( \bigl\Vert g^{j+1} \bigr\Vert _{H^{-1}(\Omega )} + \bigl\Vert f^{j+1} \bigr\Vert \bigr) \Biggr)^{2}. \end{aligned} $$
    (15)

To obtain the error estimate between the solutions u of (1) and \({(u)^{k}}_{0\leq k\leq K}\) of (5), we define the error ϒ k =( e ( u ) k e ( v ) k ) such that \(e(u)^{k}=u(t_{k}) - u^{k}\) and \(e(v)^{k} =v(t_{k}) - v^{k}\). We can easily show that \({(\Upsilon )^{k}}_{0\leq k\leq K}\) is a solution of (13), where the two components of \(F^{k+1}\) are the following consistency errors:

$$ \varepsilon (u)^{k}=\frac{u(t_{k+1})-u(t_{k})}{\delta t_{k}}- \partial _{t} u(t_{k+1}), \qquad \varepsilon (v)^{k}= \frac{v(t_{k+1})-v(t_{k})}{\delta t_{k}}-\partial _{t}v(t_{k+1}). $$
(16)

Theorem 1

If a solution u of problem (1) belongs to \(W^{3,1}(0,T;L^{2}(\Omega ))\cap W^{2,1}(0,T;H^{1}_{0}(\Omega ))\), then

$$ \begin{aligned} &\bigl\Vert \varepsilon (u)^{k} \bigr\Vert ^{2} + \bigl\Vert \nabla \bigl(u(t_{k}) - u^{k}\bigr) \bigr\Vert ^{2} \\ &\quad \leq C \delta t^{2} \biggl( \int _{0}^{t_{k}}\bigl( \bigl\Vert \partial ^{3}_{t}u \bigr\Vert + \bigl\Vert \partial ^{2}_{t} \nabla u \bigr\Vert \bigr) (s)\,ds \biggr)^{2},\quad {0\leq k\leq K}, \end{aligned} $$
(17)

where C is a positive constant independent of δt.

Proof 3

Since \({(\Upsilon )^{k}}_{0\leq k\leq K}\) is a solution of (13), where the second member is \(F^{k+1}\). Then applying the stability condition (14) leads to

$$ \begin{aligned} &\bigl\Vert e(u)^{k} \bigr\Vert ^{2} + \bigl\Vert \nabla e(u)^{k} \bigr\Vert ^{2} \\ &\quad \leq 2 \bigl( \bigl\Vert e(v)^{0} \bigr\Vert ^{2} + \bigl\Vert \nabla e(u)^{0} \bigr\Vert ^{2} \bigr) +2 \Biggl(\sum_{j=1}^{k} \delta t_{j} \bigl( \bigl\Vert \varepsilon (v)^{j} \bigr\Vert + \bigl\Vert \nabla \varepsilon (u)^{j} \bigr\Vert \bigr) \Biggr)^{2} \end{aligned} $$
(18)

thanks to the Taylor theorem with integral remainder to bound the terms \(\| \varepsilon (v)^{j} \| \), \(\| \nabla \varepsilon (u)^{j}\| \), \(\| e(v)^{0} \| \), and \(\| \nabla e(u)^{0} \| \). Then we conclude the desired estimate (17).

We can find the following error estimate in weaker norms by using the same technique as the proof of Theorem 1 and replacing condition (14) by (15).

Corollary 1

Suppose that the solution u of system (1) belongs to \(W^{3,1}(0,T;L^{2}(\Omega ))\cap W^{2,1}(0,T;H^{1}_{0}(\Omega ))\). Then the following a priori error estimate between the solution u and the solution \({(u)^{k}}_{0\leq k\leq K}\) of system (5) holds for \({0\leq k\leq K}\):

$$ \bigl\Vert \varepsilon (u)^{k} \bigr\Vert _{H^{-1}(\Omega )}^{2} + \bigl\Vert \nabla \bigl(u(t_{k}) - u^{k}\bigr) \bigr\Vert ^{2}\leq C \delta t^{2} \biggl( \int _{0}^{t_{k}}\bigl( \bigl\Vert \partial ^{3}_{t}u \bigr\Vert _{H^{-1}(\Omega )} + \bigl\Vert \partial ^{2}_{t} \nabla u \bigr\Vert \bigr) (s)\,ds \biggr)^{2}, $$
(19)

where C is a positive constant independent of δt.

We remark that the obtained estimates (18) and (19) are optimal of order 1 in time, since the discretization is based on the implicit Euler scheme, which is of order 1.

The spectral discretization

We further suppose that Ω is a rectangle for \(d=2\) or a parallelepiped rectangle for \(d=3\).

Let \(\mathbb{P}_{N}(\Omega )\) the space of polynomials of degree ≤N (\(N\geq 2\)) for each variable, and let \(\mathbb{P}_{N}^{0}(\Omega )= \mathbb{P}_{N}(\Omega )\cap H^{1}_{0}( \Omega )\). We define \(\zeta _{i}\), \(0\le i \le N\), the set of nodes, roots of the polynomial \((1-x^{2})L_{N}'\), where \(L_{N}\) is the Legendre polynomial, and \(\varrho _{i}\), \(0\le i \le N\), are the weight set of the following Gauss–Lobatto quadrature formula on the interval \(]{-}1,1[\):

$$ \forall \eta _{N} \in \mathbb{P}_{2N-1} \bigl(]{-}1,1[\bigr),\quad \int _{-1}^{1} \eta _{N}(x) \,dx=\sum _{i=0}^{N} \eta _{N}(\zeta _{i})\varrho _{i}. $$
(20)

We recall the following property (see [10, 13]):

$$ \forall \eta _{N} \in \mathbb{P}_{N} \bigl(]{-}1,1[\bigr), \quad \Vert \eta _{N} \Vert _{L^{2}(]{-}1,1[)}^{2} \le \sum_{i=0}^{N} \eta _{N}^{2}(\zeta _{i}) \varrho _{i} \le 3 \Vert \eta _{N} \Vert _{L^{2}]{-}1,1[}^{2}. $$
(21)

The reference domain \(]{-}1,1[^{d}\) (\(d=2, 3\)) is transformed to the domain Ω using the affine mapping T, and the scalar product is defined on continuous functions u and v by

$$ \begin{aligned} &(u,v)_{N} \\ &\quad = \textstyle\begin{cases} \frac{\operatorname{meas}(\Omega )}{4} \sum_{i=0}^{N} \sum_{j=0}^{N} (u \circ T)(\zeta _{i},\zeta _{j}) (v \circ T)(\zeta _{i}, \zeta _{j}) \varrho _{i} \varrho _{j} & \text{if } d=2, \\ \frac{\operatorname{meas}(\Omega )}{8} \sum_{i=0}^{N} \sum_{j=0}^{N} \sum_{k=0}^{N} (u \circ T)(\zeta _{i},\zeta _{j}, \zeta _{k}) (v \circ T)(\zeta _{i},\zeta _{j},\zeta _{k}) \varrho _{i} \varrho _{j} \varrho _{k} & \text{if } d=3. \end{cases}\displaystyle \end{aligned} $$
(22)

Remark 3

For simplicity of analysis, we suppose that the spectral discretization is fixed over time.

We suppose that \(u_{0}\) and \(v_{0}\) are continuous on Ω̄. The discrete problem is deduced from (5) by applying the Galerkin method combined with numerical integration:

$$ \text{For } u_{N}^{0}= \mathfrak{I}_{N}(u_{0}) \quad \text{and}\quad u_{N}^{1}= \mathfrak{I}_{N}(u_{0})+ \delta t_{0}\mathfrak{I}_{N}(v_{0}) \quad \text{in } \Omega , $$
(23)

where \(\mathfrak{I}_{N}\) is the interpolating operator from \(L^{2}(\Omega )\) into \(\mathbb{P}_{N}(\Omega )\), find \(u_{N}^{k}\in \mathbb{P}_{N}^{0}(\Omega )\times \mathbb{P}_{N}( \Omega ) \times (\mathbb{P}_{N}^{0}(\Omega ))^{K-1} \), \(1\le k \le K\), such that for all \(v_{N} \in \mathbb{P}_{N}^{0}(\Omega )\),

$$ \biggl(\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} - \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}},v_{N} \biggr)_{N}+ \delta t_{k}\bigl( \nabla u_{N}^{k+1}, \nabla v_{N} \bigr)_{N} = 0. $$
(24)

As in (6), \(u_{N}^{k+1}\), \(1\le k \le K\), is the solution of the discrete weak problem

$$ \bigl(u_{N}^{k+1}, v_{N} \bigr)_{N} + \delta t_{k}^{2} \bigl(\nabla u_{N}^{k+1}, \nabla v_{N} \bigr)_{N}= \biggl(u_{N}^{k} + \frac{\delta t_{k}}{\delta t_{k-1}}\bigl(u_{N}^{k}-u_{N}^{k-1} \bigr),v_{N}\biggr)_{N}. $$
(25)

Proposition 3

Let the data \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\). If \(u_{N}^{0}\) and \(v_{N}^{0}\) are known, then problem (25) has a unique solution \(u_{N}^{k+1}\), \(k\geq 1\), in \(H^{1}_{0}(\Omega )\). Moreover, the solution \({(u_{N}^{k})}_{0\leq k\leq K}\) of problem (23)(24) satisfies for \(0\leq k\leq K\) the following stability condition:

$$ \begin{aligned} &\biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert ^{2} \\ &\quad \leq {\bigl(3^{d}\bigr)}^{K} \bigl( \bigl\Vert \mathfrak{I}_{N}(v_{0}) \bigr\Vert ^{2} +2 \bigl\Vert \nabla \mathfrak{I}_{N}(u_{0}) \bigr\Vert ^{2} +2\delta t_{0}^{2} \bigl\Vert \nabla \mathfrak{I}_{N}(v_{0}) \bigr\Vert ^{2} \bigr). \end{aligned} $$
(26)

Proof 4

We show that problem (25) has a unique solution using the Lax–Milgram theorem and property (21).

To prove the stability condition (26), we define \(\|\cdot \|_{d}\) the discrete norm deduced from the discrete scalar product \((\cdot ,\cdot )_{N}\). Now letting \(v_{N}=\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}}\) in (24) leads to

$$ \biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert _{d}^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert _{d}^{2}=\biggl(\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}}, \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}}\biggr)_{N}+\bigl(\nabla u_{N}^{k+1}, \nabla u_{N}^{k}\bigr)_{N}. $$

Using the Cauchy–Schwarz inequality and (21), we have

$$ \biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert ^{2} \leq {3^{d}} \biggl( \biggl\Vert \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k} \bigr\Vert ^{2} \biggr). $$

Then iterating over k, we obtain

$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq { \bigl(3^{d}\bigr)}^{K} \biggl( \biggl\Vert \frac{u^{1}-u^{0}}{\delta t_{0}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{1} \bigr\Vert ^{2} \biggr). $$

Finally, estimate (26) is deduced from (23).

Proposition 4

Let \(u_{0}\), \(v_{0}\) be continuous on Ω̅, and let \(u_{N}^{0}\), \(v_{N}^{0}\) be known. The error estimate between solutions \(u^{k+1}, k\geq 1\), and \(u_{N}^{k+1}\), \(k\geq 1\), of problems (6) and (25), respectively, is

$$ \begin{aligned} \bigl\Vert u^{k+1} - u_{N}^{k+1} \bigr\Vert & \le C \Biggl(\inf _{\chi _{N}^{k+1} \in \mathbb{P}_{N}^{0}(\Omega )} \bigl\Vert u^{k+1}-\chi _{N}^{k+1} \bigr\Vert + \Biggl[ \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert + \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert \\ &\quad {} +\sum_{j=1}^{k} \bigl(T^{1,j}+T^{2,j}+T^{3,j}\bigr) \Biggr] \Biggr), \end{aligned} $$
(27)

where

$$ \begin{aligned} &T^{1,j} = {\frac{1}{\delta t^{2}_{j}}} \sup _{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }(u^{j+1}-u^{j})v_{N} \,d{\mathbf{x}} -(\chi _{N}^{j+1}-\chi _{N}^{j},v_{N})_{N}}{ \Vert v_{N} \Vert }, \\ &T^{2,j} = \sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- (\nabla \chi _{N}^{j+1},\nabla v_{N})_{N}}{ \Vert v_{N} \Vert }, \\ &T^{3,j} = \sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }(u^{j}-u^{j-1}) v_{N} \,d{\mathbf{x}} - (\mathfrak{I}_{N}(u^{j}-u^{j-1}) , v_{N} )_{N}}{ \Vert v_{N} \Vert }, \end{aligned} $$

and C is a positive constant independent of N.

Proof 5

Consider \(\chi _{N}^{k+1}\in \mathbb{P}_{N}^{0}(\Omega )\). By the triangle inequality we have

$$ \bigl\Vert u^{k+1} - u_{N}^{k+1} \bigr\Vert \le \bigl\Vert u^{k+1} - \chi _{N}^{k+1} \bigr\Vert + \bigl\Vert \chi _{N}^{k+1} - u_{N}^{k+1} \bigr\Vert . $$

To estimate \(\|u_{N}^{k+1}- \chi _{N}^{k+1}\|\), we begin by writing problems (5) and (25) for \(v_{N}\in \mathbb{P}_{N}^{0}(\Omega )\). Then we consider \(\tau _{k} = \frac{\delta t_{k}}{\delta t_{k-1}}\) and doing the difference term by term, we obtain

$$ \bigl(u_{N}^{k+1}-\chi _{N}^{k+1},v_{N} \bigr)_{N}+ \delta t_{k}^{2}\bigl(\nabla \bigl(u_{N}^{k+1}- \chi _{N}^{k+1} \bigr),\nabla v_{N}\bigr)_{N}=\bigl(u_{N}^{k}- \chi _{N}^{k},v_{N}\bigr)_{N} + \tau _{k} \mathcalligra{K}^{k} (v_{N}), $$

where

$$ \begin{aligned} \mathcalligra{K}^{k} (v_{N})={}& \frac{1}{\delta t^{2}_{k}} \biggl( \int _{\Omega }\bigl(u^{k+1}-u^{k} \bigr)v_{N} \,d{\mathbf{x}} -\bigl(\chi _{N}^{k+1}- \chi _{N}^{k},v_{N}\bigr)_{N} \biggr) \\ &{} + \int _{\Omega }\nabla u^{k+1}\nabla v_{N} \,d{\mathbf{x}}- \bigl(\nabla \chi _{N}^{k+1},\nabla v_{N}\bigr)_{N} \\ &{} + \int _{\Omega }\bigl(u^{k}-u^{k-1}\bigr) v_{N} \,d{\mathbf{x}} - \bigl(\mathfrak{I}_{N} \bigl(u^{k}-u^{k-1}\bigr) , v_{N} \bigr)_{N}. \end{aligned} $$

Since \(\mathcalligra{K}^{k}\) is linear and continuous on \(\mathbb{P}_{N}^{0}(\Omega )\), by the Riesz theorem there exists \(\vartheta ^{k}_{N}\) in \(\mathbb{P}_{N}^{0}(\Omega )\) such that

$$ \mathcalligra{K}^{k} (v_{N})=\bigl(\vartheta ^{k}_{N},v_{N}\bigr)_{N}. $$

Applying the result proved in [14, Prop. 4.1] and [15], we get

$$ \bigl\Vert u_{N}^{k+1}- \chi _{N}^{k+1} \bigr\Vert \le C \Biggl( \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert + \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert + \sum_{j=1}^{k} \bigl\Vert \vartheta ^{j}_{N} \bigr\Vert ^{2} \Biggr)^{1/2}, $$

where C is a positive constant independent of N.

So we conclude (27), since

$$ \bigl\Vert \vartheta ^{j}_{N} \bigr\Vert \leq C \sup_{v_{N}\in \mathbb{P}_{N}^{0}( \Omega )} \frac{(\vartheta ^{j}_{N},v_{N})_{N}}{ \Vert v_{N} \Vert },$$

where C is a positive constant independent of N.

To find the order of convergence as a function of N, it is necessary to estimate each of the terms of the second member of inequality (27).

  • Estimation of \(T^{1,j}\)

We consider \(\varpi ^{j+1}=u^{j+1} - u^{j}\) and \(\chi _{N}^{j+1}-\chi _{N}^{j}= \Pi ^{1,0}_{N-1}(\varpi ^{j+1})\). By the exactness of the Gauss–Lobatto quadrature formula of (20), \(\int _{\Omega }\Pi ^{1,0}_{N-1}(\varpi ^{j+1}) v_{N} \,d{\mathbf{x}}\) and \((\Pi ^{1,0}_{N-1}(\varpi ^{j+1}),v_{N})_{N}\) are equal, and thus

$$ T^{1,j} \leq \bigl\Vert \varpi ^{j} - \Pi ^{1,0}_{N-1}\bigl(\varpi ^{j}\bigr) \bigr\Vert , $$
(28)

where \(\Pi _{N}^{1,0}\) is the orthogonal projection operator from \(H^{1}_{0}(\Omega )\) into \(\mathbb{P}_{N}^{0}(\Omega )\) related to the inner product defined by the semi norm \(| \cdot | _{1,\Omega }\). (See ([13], Lemma VI.2.5) and [10] for all the properties of this operator.)

  • Estimation of \(T^{2,j}\)

Since the Gauss–Lobatto quadrature formula is exact for a polynomial of degree \(\leq 2N-1\), we have

$$ \begin{aligned} &\int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- \bigl(\nabla \chi _{N}^{j+1},\nabla v_{N}\bigr)_{N} \\ &\quad = \int _{\Omega } \nabla \bigl(u^{j+1}-\Pi _{N -1}^{1,0}u^{j+1}\bigr) \nabla v_{N} \,d{\mathbf{x}} - \bigl(\nabla \bigl(\chi _{N}^{j+1}-\Pi _{N -1}^{1,0} \chi _{N}^{j+1}\bigr), \nabla v_{N} \bigr)_{N}. \end{aligned} $$
(29)

Thanks to the triangle and Cauchy–Schwarz inequalities, we have

$$ \begin{aligned} &\sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- (\nabla \chi _{N}^{j+1},\nabla v_{N})_{N}}{ \Vert v_{N} \Vert } \\ &\quad \le \bigl( \bigl\vert u^{j+1}-\Pi _{N -1}^{1,0}u^{j+1} \bigr\vert _{1,\Omega } + \bigl\vert \chi _{N}^{j+1}-\Pi _{N -1}^{1,0} \chi _{N}^{j+1} \bigr\vert _{1,\Omega } \bigr). \end{aligned} $$
(30)

Thus we conclude using the properties of \(\Pi _{N-1}^{1,0}\).

  • Estimation of \(T^{3,j}\)

Let \(\theta ^{j}=u^{j}-u^{j-1}\). We use for this estimation \(\Pi _{N -1}\) the orthogonal projection from \(L^{2}(\Omega )\) into \(\mathbb{P}_{N-1}(\Omega )\). By the same argument above for the Gauss–Lobatto formula, we have

$$ \begin{aligned} &\int _{\Omega }\theta ^{j}({\mathbf{x}})v_{N}({ \mathbf{x}}) \,d{\mathbf{x}} - \bigl( \mathfrak{I}_{N} \theta ^{j},v_{N}\bigr)_{N} \\ &\quad = \int _{\Omega }\bigl(\theta ^{j}- \Pi _{N-1} \theta ^{j}\bigr) ({\mathbf{x}}) v_{N} ({\mathbf{x}}) \,d{\mathbf{x}} -\bigl( \mathfrak{I}_{N} \theta ^{j} - \Pi _{N -1} \theta ^{j}, v_{N}\bigr)_{N}. \end{aligned} $$

Using inequality (21) in each direction leads to

$$ \int _{\Omega }\theta ^{j}({\mathbf{x}})v_{N}({ \mathbf{x}}) \,d{\mathbf{x}} - \bigl( \mathfrak{I}_{N}\theta ^{j},v_{N}\bigr)_{N}\le \bigl[ \bigl\Vert \theta ^{j}-\Pi _{N -1} \theta ^{j} \bigr\Vert ^{2} + 9 \bigl\Vert \theta ^{j}- \mathfrak{I}_{N} \theta ^{j} \bigr\Vert ^{2} \bigr] \Vert v_{N} \Vert . $$

Thanks to the approximation properties of operator \(\Pi _{N -1}\) (see [10, Thm. 7.1]) and \(\mathfrak{I}_{N}\) (see [10, Thm. 14.2]), for \(\theta ^{j}\in H^{s} (\Omega )\); \(s>1\), we obtain

$$ \sup_{v_{N}\in \mathbb{P}_{N}(\Omega )} \frac{ \int _{\Omega } \theta ^{j}({\mathbf{x}})v_{N}({\mathbf{x}}) \,d{\mathbf{x}} - (\theta ^{j},v_{N})_{N}}{ \Vert v_{N} \Vert } \le C N^{-2s} \bigl\Vert \theta ^{j} \bigr\Vert ^{2}_{s,\Omega } . $$
(31)

Finally, to estimate

$$ \inf_{\chi _{N}^{k+1}\in \mathbb{P}_{N}^{0}(\Omega )} \bigl\Vert u^{k+1}- \chi _{N}^{k+1} \bigr\Vert , \qquad \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert \quad \text{and} \quad \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert , $$
(32)

we choose, respectively, \(\chi _{N}^{k+1}=\Pi _{N}^{1,0}u^{k+1}\), \(u_{N}^{0}=\Pi _{N}^{1,0}u_{0}\), and \(v_{N}^{0}=\Pi _{N} v_{0}\). Then we conclude using properties of operators \(\Pi _{N}^{1,0}\) and \(\Pi _{N}\).

So, from estimates (28), (30), (31), and (32) we obtain the following main theorem.

Theorem 2

For \((u_{0},v_{0})\) continuous on Ω̄, solution \((u^{k})_{0\leq k\leq K}\) of problem (5) belongs to \(H^{s}(\Omega )\); \(s> 1\). The error between solutions \(u^{k+1}\) and \((u_{N}^{k+1})\) of problems (6) and (25), respectively, satisfies

$$ \begin{aligned} \bigl\Vert u^{k+1}-u_{N}^{k+1} \bigr\Vert \le{}& C \Biggl[N^{-s} \Biggl( \bigl\Vert u^{k+1} \bigr\Vert _{s,\Omega }+ \sum _{j=1}^{k} \bigl(\delta t^{-2}_{j} \bigl\Vert u^{j+1}-u^{j} \bigr\Vert _{s,\Omega } + \bigl\Vert u^{j}-u^{j-1} \bigr\Vert _{s,\Omega } \bigr) \Biggr) \\ &{} + N^{1-s} \sum_{j=1}^{k} \bigl\Vert u^{j+1} \bigr\Vert _{s,\Omega } \Biggr], \end{aligned} $$
(33)

where C is a positive constant independent of N.

Numerical results

Consider the interpolating Lagrange polynomial \(\varphi _{j}\) defined by

$$ \varphi _{j}\in \mathbb{P}\bigl([-1,1]\bigr),\quad \varphi _{j}(\zeta _{i})= \delta _{ij},\quad 0\leq i,j \leq N, $$

where \(\delta _{ij}\) is the Kronecker symbol. The solution \(u_{N}^{k+1}\) of problem (25) is written as

$$ u_{N}^{k+1}(x,y)=\sum_{i=1}^{N-1} \sum_{j=1}^{N-1}u_{N}^{k+1}( \zeta _{i}, \zeta _{j}) \varphi _{i}(x) \varphi _{j}(y). $$

Let \(U^{k+1}\) be the admissible solution vector of components \(u_{N}^{k+1}(\zeta _{i},\zeta _{j})\). The matrix system deduced form of the discrete problem (25) is written

$$ \bigl(D^{k+1}+\delta t_{k}^{2} A^{k+1}\bigr) U^{k+1}=F^{k}, $$
(34)

where \(D^{k+1}\) is a diagonal matrix of coefficients \(\varrho _{r} \varrho _{s}\), \(1\le r, s \le N-1\), \(A^{k+1}\) is the matrix with coefficients \((\nabla (\varphi _{i} \varphi _{j});\nabla (\varphi _{r} \varphi _{s}))\), \(1 \le i,j,r, s\le N-1\), and \(F^{k}\) is the vector with components

$$ u_{N}^{k}(\zeta _{r},\zeta _{s})+\frac{\delta t_{k}}{\delta t_{k-1}} \bigl(u_{N}^{k}( \zeta _{r},\zeta _{s})- u_{N}^{k-1}( \zeta _{r},\zeta _{s}) \bigr)\varrho _{r} \varrho _{s}, \quad 1 \le r,s\le N-1. $$

We remark that the matrix \(D^{k+1}+\delta t_{k}^{2} A^{k+1}\) is symmetric and positive definite. Then system (34) is solved using the gradient conjugate method at each iteration.

Iterative algorithm

Step 1: Let

$$ u_{N}^{0}= \mathfrak{I}_{N}(u_{0}) \quad \text{and} \quad u_{N}^{1}= \mathfrak{I}_{N}(u_{0})+ \delta t_{0}\mathfrak{I}_{N}(v_{0}). $$

Step 2: Suppose \(u_{N}^{k-1}\) and \(u_{N}^{k}\) are known The linear system (34) is solved by the gradient conjugate method

$$ \bigl(D^{k+1}+\delta t_{k}^{2} A^{k+1}\bigr) U^{k+1}=F^{k}. $$

We do the iterations until the following condition is satisfied:

$$ \bigl\Vert U_{N}^{k+1} - U_{N}^{k} \bigr\Vert _{H^{1}(\Omega )}\leq \xi , $$

where \(\xi =10^{-10}\) for all the following numeric tests.

Time convergence

We consider the domain \(\Omega =\,]{-}1,1[^{2}\). Two exact solutions are tested.

The first one is

$$ u(t,x,y)=e^{t} \sin (\pi x) \sin (\pi y). $$
(35)

We choose \(T=1\), \(N=20\), and \(\delta t=10^{-k}\), \(k=1, \ldots, 4\). Figure 1 deals with the quantities \(\log _{10}\|u-u_{N\delta t}\|_{H^{1}(\Omega )}\) (in blue) and \(\log _{10}\|u-u_{N\delta t}\|_{L^{2}(\Omega )}\) (in red) as functions of \(\log _{10}(\delta t)\).

Figure 1
figure 1

Time error for solution (35)

For test 2, we study the singular solution

$$ u(t,x,y)=t^{3/2}\bigl(1-x^{2} \bigr)^{5/2}\bigl(1-y^{2}\bigr)^{5/2}. $$
(36)

Figure 2 deals with the same curves as in Fig. 1 tested for the solution (36) when \(N=20\), \(T=0.1\), and \(\delta t=5. 10^{-2}, 10^{-2}, 5. 10^{-3}, 10^{-4}\). The obtained results show the convergence of the method with an order of convergence almost equal to 1.

Figure 2
figure 2

Time error for solution (36)

Spectral convergence

In this test, we fix \(\delta t=0.01\) and take \(N=5, 7, 9, 12, 14, 15, 17, 18, 20, 22\) and \(T=1\). We consider

$$ u=(1+t) \bigl(1-x^{2}\bigr) \bigl(1-y^{2} \bigr). $$
(37)

Figure 3 (respectively, Fig. 4) deals with \(\log _{10}\|u-u_{N\delta t}\|_{H^{1}(\Omega )}\) (in blue) and \(\log _{10}\|u-u_{N\delta t}\|_{L^{2}(\Omega )}\) (in red) as functions of N (respectively, \(\log _{10}(N)\)). We remark that the error norms \(\log _{10}\|u-u_{N\delta t}\|_{H^{1}(\Omega )}\) decrease exponentially until \(N=10\) and stagnate for \(N>10\). The errors \(\log _{10}\|u-u_{N\delta t}\|_{L^{2}(\Omega )}\) decrease until \(N=10\) and stagnate for \(N>10\). We remark that convergence stagnates. This is due to the fact that the time order of convergence is less than the order of the spectral method.

Figure 3
figure 3

Semilogarithmic spectral error for solution (37)

Figure 4
figure 4

Logarithmic spectral error for solution (37)

Finally, the isovalues of the exact and discrete solutions (37) are presented in Fig. 5.

Figure 5
figure 5

Spectral convergence: Continuous solution (right), discrete solution (left)

Conclusion and future work

This work concerns the numerical analysis of the implicit Euler scheme in time and the spectral discretization in space of the second-order wave equation. We prove an optimal error estimate in time and space. These estimates depend only on the regularity of the solution. Although the spectral methods are known as high-order methods in space, we remark that they have the disadvantage of losing part of this accuracy due to lower order of temporal discretization (often of order 1 or 2). The numerical analysis and implementation of the more difficult case where the spectral discretization depends on time using the second-order BDF method for the time discretization will be the subject of our forthcoming work.

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Researchers Supporting Project number (RSP-2021/153), King Saud University, Riyadh, Saudi Arabia.

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Abdelwahed, M., Chorfi, N. Numerical resolution of the wave equation using the spectral method. Bound Value Probl 2022, 15 (2022). https://doi.org/10.1186/s13661-022-01601-5

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Keywords

  • Wave equation
  • Spectral method
  • Backward Euler scheme
  • A priori error estimate
  • Implementation